Google
This is a digital copy of a book that was preserved for generations on Hbrary shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for informing people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at |http : //books . google . com/|
ENGLISH SCHOOL-CLASSICS
Edited by FRANCIS STORR, M.A.,
By tht Rev. J, F. Bbioht. h.
COWPER'S TASK.
Bf FiAHCis Stoib, M.A. u. Tait I. (Uookl.— The Sofi; Book II.-
The Timepiece) gd. Part II, {Book 111.— The Gaidcu; Hook IV.— The
Winter Kveniog) ad. Part HI. (Book V.-The Winler Morning Walk;
Book VL— The Wicier Walk at Noon) 9,/.
SCOTT'S LAY OF THE LAST HIMSTREL.
By J. SuKTBBS Phii.ltotts, M.A^ Hcid-MasiiT of Bedford Granimai
SCOTT-S LADY OP THE LAKE.
By R. W. Taylor, M.A.. Head-MutQ of Kelly College, Tavistock, u.;
or in Three Pans Q^ each.
NOTES TO SCOTT'S WAVER LEY.
By H. W. Eve, hi. A, Head-hlister oT UiilTersiiy Collees School, London.
It.; WAVEKtKV AHD NOTFS, aj. 6<L
TWENTY OF BACON'S ESSAYS.
BjrPftAHCU StOKI, M.A II.
SIMPLE POEMS.
By W. E. UuLLiNi, M.A, Ai^iium-Muler at MarlborouGh College- id.
SELECTIONS FROM WORDSWORTH'S POEMS.
ByH. H. TUHMKS, B.A, late Scholar of Trinity CoUcBe, Camhridge. a.
WORDSWORTH'S EXCURSION; The Wanderet.
By H. H. TusNM, B.A .j.
MILTON'S PARADISE LOST.
By Pkahcii Stohk. hl.A. Book I. gd!. Book II, 9^
MILTON-S L'ALLEGRO, IL PENSEROSO, AND LYCIDAS.
By EnwABD Stobh, JI.A., late Seholii of New ColleRe, Oxford. 11.
SELECTIONS FROM THE SPECTATOR.
By OiHUHn Am, U.A, late Aubtanl- Muter at Wellington Colleie. 11.
BROWNE'S RBLIOIO MEDICI.
By W, P. Shith, M.A., Auistant-Mailer at Winchester College, m.
GOLDSMITH'S TKAVELLER AND DESERTED VILLAGE.
By C Sahkiv, MJL, Assistant-Master at Mirlborough Collete. u.
EXTRACTS fmm GOLDSMITH'S VICAR OF WAKEFIELD.
ByC.SAHKav, M.A. u.
POEMS SELECTED from the WORKS OF ROBERT BURNS.
ByAM. Buj.,M.A, BaUielCoUege, Oifoid. li.
HACAULAY'B ESSAYS:
^Qjj By F.AKC.5 Stohii. M.A „
H.A 9d.
MpORKJS LIFE^OF^BYRON^^BjFbahcis Stobi^^M.A *t
SchoLu of BraunoM College, OiTord. i
SOUTHEY'S LIFE OF NELSON.
By W. E. MtiLLtNS, U.A ». 6d.
GRAY'S POEMS with JOHNSON'S LIFE AND SELECTIONS
from GRAY'S LETTERS. By Fbaheis Stoib. M.A u.
Waterloo Place, Pall Mall, London.
RIVINGTONS' MATHEMATICAL SERIES
^ ■ «. • ^/»
By J. HAMBLIN SMITH, M.A.,
OP GONVILLB ANU CAIUS CUI.LKGR, AND LATR LECTURF.R AT ST. I'BTRR'S COI.T.imR,
CAMnKIIXSB.
Arithmetic. 3^. dd, a key, 91.
Algebra, part r. 3^. without Answers, 2j. 6</. A Key, 9^.
Exercises on Algebra. Part i. 2.r. 6/.
[Copies may be had without the Answers.]
Elementary Trigo7iometry , 4^. 6./. a key, 71. 6^/.
Elements of Geometry.
Containing Books I to 5, and portions of Books ii and 12 of
Euclid, with Exercises and Notes, 3J. 6</. A Key, %s, td.
PART I., containing IJooks I and 2 of EucT.iD, may be had
separately.
Elefnentary Hydrostatics, v^ \
. [a Key, 6f.
Elementary Statics, 3^. )
Book of Entmciations
FOR IIamblin Smith's Geometry, Ai.orbra, Trigono-
metry, Statics, and Hydrostatics, ij.
T/ie Sttidy of Heat, y.
By E. J. GROSS, M.A.,
FRLI.OW OP GONVILLB AND CAIUS COLLBGK, CAMBRIIXIR, AND SRCRRTAKY TO
THK OXFORD AND CAMBRIDGE SCHOOLS EXAMINATION BOARD.
Algebra. Part 11. 8j. (hU
Kinematics and Kinetics, sj. u.
By G. RICHARDSON, M.A.,
ASSISTANT-MASTKK AT WINCHKSTKR COLLEGE, AND LATE FRLI.OW OF ST. JOHN'S
COLLEGE, CAMBRIDGE.
Geometrical Conic Sections. ^. u.
^^^^^^«^^^^rf^ N » • V^fiV^ -^iCS^ -^ ^ •^" • *.«. «..*,■« «>,«^^k^.^ • ^••^M -»^ - • * » •
Waterloo Place, Pall Mall, London.
2
A KEY
TO
ELEMENTARY STATICS AND
HYDROSTATICS
-1
A KEY
TO
ELEMENTARY STATICS AND
HYDROSTATICS
BY
M
J. HAMBLIN SMITH, M.A.
OP GONVILLB AND CAIUS COLLRGB,
AND LATE LECTURER AT ST. PETEK'S COLLEGE, CAMURIDGK
M » *J V w
RIVINGTONS
WATERLOO PLACE, LONDON
MDCCCLXXXIII
ELEMENTARY STATICS.
KEY.
Examples — I. (p. 25).
1. Let C be the body. ACB a straight line.
Then if the three forces act all in the same direction along CBy a
force of 12 lbs. will be required acting along CA to keep them at rest.
^
c
B
3
4-
5
A
c
B
3
5
4
A
^
c
B
4
\
7
A_
^
c
— ^—
^
B
Fio. 1.
If the forces of 5 lbs. and 4 lbs. act along CB^ and the force of 3 lbs.
along C-4, a force of 6 lbs. wiU be required acting along CA to keep
equilibrium ; and so for the other cases.
2. As in the preceding example, if the forces all act along OB they
will have a resultant acting along CB of 29 lbs.
If the forces of II lbs. and 13 lbs. act along CB^ and the force of 5
lbs. along CA^ thej will have a resultant of 19 lbs. acting along CB ;
and so for the other cases.
3. Let X be the measure of the magnitude of the resultant in lbs.
Then a2= (90)2 + (120)2
=8100 + 14400=22500;
.'. x=I50, and the resultant is one of 150 lbs.
A
KEY TO ELEMENTARY STATICS.
4. Here «*= (36)2 + (48)2
= 1296 + 2304 = 3600;
.'. «= 60, and the resultant is one of 60 lbs.
5. As the squares of the numbers 6 and 8, that is, 36 and 64, are
together equal to 100, which is the square of 10, it is plain from the
first diagram on page 23 of the Statics, that if the measures of AC,
AB be 8 and 6, and the measure of AD be 10, the square on AD
will be equal to the sum of the squares on AC, CD, and, therefore,
I ACD will be a right angle (Euclid, I. xlyiii) ; whence i BAC is
also a right angle (Euclid, I. xxix).
6. Since the squares of 3 and 4, that is, 9 and 16, are together
equal to 25, which is the square of 5, the forces 3 and 4 will act at
f^ jQ right angles to each other. Hence if C
be the point, draw CA, CB containing
3 and 4 units of length respectively,
Q complete the parallelogram ACBD,
Produce DC to E, making CE^DC^b
units. Then GA, CB, CE will repre-
£ sent the arrangement of the three
forces.
FiO. 2.
7. Let the measures of the forces be x and ,J3 . x, which numbers
are in the ratio of 1 : jj3.
Then a;« + (V3.a;)2= (10)2;
or, at^ + Zx^^lOO ; or, 4a^=100 ; or, a;=5 ;
.'. the forces are 5 lbs. and 5 ,J3 lbs.
8. Construct a diagram as in Example 2 on p. 23, but with
I BAC-^20\
Then 1 2)0^=30% L CDE^^% and CD^WE.
Hence C^=- CD^ - DE^
^CT^ Giy'.^CD^.
^^ "T" 4 '
KEY TO ELEMENTARY STATICS,
Then if a; b^ the measure of AI>^
-v/3.6
2
=9 + 25 + 15 V3=34+ 15V3 ;
.*. «=V34+15V3.
Or, by Trigonometry,
x2:«32^.5« + 2 .3.5. co830'=9 + 25 +30 . ^=34 + 16^3.
9. Construct a diagram as in Example 2 on p. 23.
Then x2=102 + 7^ + 2 . 10 . -^
2
= 100 + 49 + 70=«219;
.-. a;«V219.
Or, by Trigonometry,
a;«=102 + 7« + 2.10.7.cos60'=100 + 49 + 140.4 = 219.
2
10. Let ^-B, AC represent the forces, l BAC being an angle of
135°.
Complete the parallelogram ABDC, draw DE perpendicular to ^C
Then z ^GD= 45* (Eucud, I. xxix.), q q
and EC=^ DE.
Also, CD^^DE^ + EC^
^2E(P,
and.-.^C=^. j,^3
Hence, if aj be the measure of -4D,
x^=AC^-^CIP-2AC,EC. (Euclid, II. xiii.)
= 112 + 92-2. 11 . -5^=121 + 81-99 V2. = 202-99 ^2 ;
.-. x= a/202 -99^2.
Or, by Trigonometry,
a;2=ll2 + 92 + 2.11.9.cosl35'
= 121+81 + 198 X (--j=)=202-99 ^2.
KEY TO ELEMENTARY STATICS.
11. Construct a diagram as in Example 2 of p. 23, but having
L BAC^46% and .*. z DCE^46°.
CD
Then, as in Example 10, (^^=—7q'
Hence, if a; be the measure of J.I>
x^=AC^ + CI>^ + 2AC,CE
5
=4*^ + 52+2.4.
V2
= 16 + 25 + 20^2=41+20^2;
.-. a:= V4I + 20 ^2.
Or, by Trigonometry,
x2=42 + 52 + 2.4.5.cos45"
= 16 + 25 + 40.-^2=41 +20 V2.
12. This Example is to be worked precisely in the same way as
Example 2 on p. 23, except that instead of 8 lbs. we must put F to
represent each of the given forces. Then the magnitude of the
resultant will be F a/3 in place of 8 ^3 lbs.
Or, by Trigonometry,
a;2=jp2 + ^ + 2i^.cos60''
= 2F^ + 2F^,^ = ZFK
13. Let AB^ AG represent the forces acting at A.
Produce BA to ^, making AE=AB.
Complete the parallelogram ABDC, AEFC,
Then, if P and Q be the forces represented by AB and AC,
and if z BAC^a, and .*. z J^^(7=180°-o, and if B and R' be the
resultants in each case,
il2=p2 + Q2 + 2PQ.cosa,
ir2=.p2+Q2-2PQ.coso;
.-. jB2+jB'2 = 2(P2+Q2).
But since DAF is a right-angled triangle,
FD^^AD^ + AF^;
.'. (2P)2=i22 + ie^
Pio. 6. .-. 2P2=2^,
which cannot be uoless P=Q,
KEY TO ELEMENTARY STATICS,
14. Let D be the point where the vertical line of direction, in
-which the weight acts, meets a line drawn from A parallel to BF,
Then the component forces, which are ¥ A
and the tension of the string BA^ being
represented by BG and BA^ BD will repre-
sent the resultant force of 10 lbs.
Now, DBF is a right angle, and z DCB
=60°. (Euclid, I. xxix.)
/.BD^^Z.BC;
...p-i»,u-!»3^5,u.
Uiolhs
Fio. 6.
Examples— 11. (p. 31).
1. As in Example 12 of I., we find the resultant of P and P to be
P . V3 ; and the resultant of Q and Q is mJq^'+Q^^ or Q . a/2~;
.'. since the resultants are equal,
P,J3--Q^2, that is, P : Q-^>J2 : V3.
2. Let P and Q be the forces, and let P, their resultant, be equal
to P. Draw a diagram as in Art 38.
Then since AB^AD, .-. z ABD= l ABB.
Now L ABD=eO% because z 5^40=120° ;
.-. z^PD and z^DP are each=60°;
.-. z Pull) ==60° (Euclid, I. xxxn.)
that is, the triangle ABD is equilateral, and /. P= Q.
3. (1) The proof depends on the fact that the diagonals of a
parallelogram bisect each other. See Art 34, 3.
Hence FB is half the diagonal of a pai^
allelogram, of which J.P, DE are the adja-
cent sides.
And EF is half the diagonal of a paral-
lelogram, of which BF, CF are the adjacent
sides.
Hence resultant of AE, DE =2PP,
and resultant of PP, CF=- 2EF ;
.', since 2PP and 2PP represent equal and opposite forces, forces
represented by AE, DE, BFy CPare in equilibrium.
Fio. 7.
KEY TO ELEMENTARY STATICS,
(2) The length of a side of the square being 5, the length of the
diagonal is V/S^ + ^S*, or ^JY.
Now the resultant of forces B and BJ^ acting at an angle of 45*
(for the diagonal bisects the angles of the square), is found, as we
proved in Example 11 of I., thus —
v"
4. If ABy iiC represent the two equal forces, and AD the third force,
Q £ z J?^C=60% because the sum of
L^BABy CAD=^ 300'.
Then, as in Example 12 of I.,
AE^^^.AB.
Hence if ^ be the magnitude of
each of the equal forces, the magni<
Pio. 8. tude of the third force is ^3 . F.
5. (1) Let be the centre of the circle described about the
triangle ABC,
A
Fio. 9.
Then OA = OB^ OC.
Now AO passes through 2>, the middle point of BC, by the con-
ditions of the question. (See Examples — II. 3 (1).)
Hence (fig. I.) OD is perpendicular to BC (Euclid, III. in.) ;
.*. AB=AC (Euclid, I. rv.), and therefore ABC is isosceles.
If 0, the centre of the circle, lies in BC, then OA^OB^OC, and
the angle BAC^ being the angle in a semicircle, is a right angle ;
that is, the triangle is ri^t-angled.
KEY TO ELEMENTARY STATICS.
7
(2) The two smaller forces being at right angles, let the measures
of the three forces, taken in order of magnitude and represented bj
AB^ ACf AD hex-dfXfX + d,
Then AE^==AC^+ CE*,
and/. {x+d)^^a^+{x-'d)^;
Hencei x^4dy and the measures of the
forces are Zd, 4d, fid ; that is, the common
difference d is one-third of the least force.
N,B. — There is an error in this Example
in the edition of 1875 : for ffreaUit read
least but one.
Pfo. 10.
6. This is worked in the same way as I. 9, and we get
«»=4 + 9 + 2.2.4- = 13 + 6-19.
7. This is worked as L 9, obsenring that the resultant is given, and
that X must be put for one of the components ; then
X
3«=2*+iB« + 2.2.
2
or, 9=4 + {B^ + 2x.
Hence «• + 2a;=5 ; whence «— ,J6 - 1.
8. Let be the point of application, OE the line pointing to the
East. O /^
Let OAy OB, represent the components,
so that Zu405=135^
Draw AR at right angles to OE.
Then since i u40iJ= 45% and i ABO B'
=90", z 0XB=45'.
.'. if the measure of AO be aJ2.P, the measure of OB=P, and
since DA = BO, the measure of DA — P.
Hence DOBA is a parallelogram.
.'. OD is parallel to A By and is therefore in the direction of the
North, and OD^AB, and therefore its magnitude is P.
8
KEY TO ELEMENTARY STATICS.
9. Let OAy OB, OC represent the three forces P, 2P, ^3 • P
acting at 0.
Complete the parallelogram OADB,
Then since OD represents 2P, and
(2P)«=( V3. Ff+F\ I DAO is a right angle ;
/.since 0D=2,0A,
zl>0^ = 60'; and.-. z^0i)=30'.
Hence z^0^=90',
z BOC^ lbO%
Pio.12. iA0C^l20\
10. Produce RO to any point 2>, and draw D^ parallel to OF,
meeting OQ in B,
Then the sides of the triangle DOB being
Q parallel to the forces P, Q, jR, are proportional
to them.
IB ^ Now since z JBOQ=136", .'. zD0^=45^
^ Also z 2)^0=90°.
••. P:Q:B=DB:BO:DO
= 1 : 1 : V2.
Fio. 13.
r^Q
no. 14.
il. Construct a diagram as in Example 10.
Then since P=C=JB,
i>JB=JB0=02>.
.-. z2>0^=:60';
.-. iROQ = l20\
Similarly it may be shown that
I FOQ= A FOR= 120\
KEY TO ELEMENTARY STATICS.
12. Let ABy ^C be the components, AD the ^
resultant perpendicular to AB,
Now, since z ^^(7=120% .*. z^JBi)=60^
Hence in triangle DAB
ABiBD'.DA^l ;2: V3;
/. AB.ACiAD^l :2: ^3.
13. Let ABy.AC represent the components, AC='2ABy and AD
the resultant.
Produce BA to E, making AE=AB,
Then since z 01^=120°, .-. i 2>^J^«60" ;
and .'., since BE^BD,
L BDE= L BED=^60°,
Hence DBE is an equilateral triangle, and DAj
which bisects the base, is at right angles to BA,
.-. AD:AB= V3 : 1, or, AD= V3 . P.
Fia. 16.
14. Let AB, ^0 be the components, and let AD the resultant
Produce AB to E, making BE==AB,
Then AE=AD.
But, since DB bisects AE at right angles, °
AD=^DE;
.'. ^^D ia an equilateral triangle, A
and .*. z ^^2) = 60'. Fio. 17.
15. Let AB, AC he the components, B
AD the resultant =3 ^C
Then since iBAC== 136', . '. z DOi = 45^
Hence z ^Z>C= z VCA=-46%
9^d,\ I D AC =90\
Then ^5 : ^C= CD\AC=^mJ2: 1.
10
KEY TO ELEMENTARY STATICS.
16. Let AO^ BO be the strings,
CO the vertical line
A From any point DiaOC draw DE parallel
E to OB.
Then the sides of the triangle DOE are
parallel to the three forces.
Now I DOE =30% iDEO^%0\
and .-. L EDO^eo\
.'. tension of -4 : tension of BO^EO: ED
= V3:1.
Fio. 19.
17. (1) Construct a diagram as in Example 13, and AD the re-
sultant, being at right angles io AB, makes an angle of 30**
with AC.
(2) The components being equal, the resultant bisects the angle
between them, and this being an angle of 135°, the resultant makes
an angle of 67^° with each component.
18. Produce AD to E, making DE^AD.
p p Join BEf CE,
Then .'. AD^ED, and BD= CD,
md I BDE=^ I ADC;
.'.iBED^iDAC.
Hence BE is parallel U> AG.
Again, •.• AD'^ED, and BD^CD,
A C and i ADB=EDC,
Fio. 20. /. Z DBA »= Z DCE.
Hence AB is parallel to EC,
Thus ABEC is a parallelogram, and AE ia the resultant of AB,
AC
KE Y TO ELEMENTAR V ST A TICS. 1 1
19. Construct a diagram as in Example 10.
Let jB, P, Q be the order of magnitude,
B, being the greatest.
Then ODy DBf BO are in descending order ;
.-. I OBD, L BOD, L ODBiae in descend-
ing order.
.-. z POQ, L QOB, I BOP are in ascend-
ing order,
for z OBD + z POQ = 2 right angles, and
similarly for the others. Fio. 2i.
20. Take a parallelogram ABCD, having z BAD less than z ABC,
Now ABD and BAC are two triangles
having two sides equal, each to each, but
the included z BAD in the one less than
the included z ABC in the other,
.'. BD is less than AC,
Or, by Trigonometry,
JJ«=pa + ^ + 2PC.co8a; Fio. 22.
and as a increases from 0^ to 90% cosa diminishes,
and as a increases from 90'' to ISO**, cosa increases, but is negative ;
.', B\a always decreased as a increases.
21. Let a be the angle between the strings, t the ten-
sion of each string, w the weight supported.
Then ti;2= e« + e* + 2«« . coso.
Now, as the string is lengthened a is decreased,
*. cosa is increased ;
and .*., since w remains constant, t is decreased. fio 28.
The pressure on the nail will not be affected by the length of the
stripg.
12
KEY TO ELEMENTARY STATICS.
22. Let A be the peg, and let AB^ AC represent the equal forces
P and P pulling at the ends of the string.
A Then the strain on the peg is a
force represented by AD, the
diagonal of the rhombus ACDB.
Also, AD-=AB=Aa
^ ,\ A CD and ABD are equilateral
D triangles ;
.-. iCAD^iBAD=eO'';
.-. iCAB=l20\
Fia. 24.
23. The polygon, the sides of which represent the forces in magni-
tude and direction, must be a rhombus or a square, for all the sides
are to be equal Hence, since the opposite sides of the polygon are
parallel, the forces must be opposite, two and two.
Fio. 25.
24. Take F, the middle point of CD, and join
EF ; then EF is parallel to ^C.
Now, resultant of AD, AC=2, AF (Example
18), and resultant of AE, AC^AF;
'\ resultant of AD, -40= twice resultant of
AE, AC.
25. In the first system B is the resultant of P and Q ; in the second
system jR' is the resultant of P and Q wheit these forces are reversed.
.*. jR is equal and opposite to B\
26. Take the diagram on page 30 of the Statics.
The force representing the combined eflfect of AB, BC, CD, DE is
AE, and therefore the forces represented by AB, BC, CD, DE are
kept in equilibrium by a force represented by a line equal and oppo-
site to AE.
Hence X lies in EA produced, so that AX—AE.
KEY TO ELEMENTARY STATICS. 13
27. Let be the point loithin the quadrilateral ABCD.
Now E, F, Gj H, being the points of bisection of the sides of the
quadrilateral, being joined would form a
parallelogram, and the diagonals of this
parallelogram bisect each other in N,
Then resultant of OA, OD is 2. 0(?,
and resultant of OB, 0Cia2,0E;
.-. resultant of OA, OB, OC, OD is
2 . resultant of OQ, OE,
But resultant of OC?, JS? is 2 . OiV ; p^^ 2^
.-. resultant of OA, OB, OC, 02) is 4. ON,
The construction and proof are precisely the same when is without
the quadrilateral.
Examples— III. (p. 40).
1. Let AB represent a force of 12 lbs., AC, AD the components.
Then z CAB^90% L ^150=30^ C B
and.-. ^-405=60^
Then AG. CB:AB=l : 2 : V3 ;
.*. AC : 12=1 : ^3, or, -4 (7 represents A ^
a force of 4 *J3 lbs ; Fio. 27.
and CB : 12=2 1^3, or, AD represents a force of 8/v/3 lbs.
2. (1) Let AB represent a force of 10 lbs., AD, AC the compo-
nents, each making an angle of 30* with AB, and being therefore
equal.
Draw BE at right angles to AC
produced.
Then i BCE^ i DAC^eO"" ;
md ,\ BC=2CE.
Ji^ow A&=^AC^+CB^ + 2AC.CE,
AC
or 100^ AC^ + AC^ + 2AC.
.-. 100=3^ C«, and .-. AC^^-^^AD.
«5
14
KE y TO ELEMENTAR V STA TICS.
(2) Since i BAD^ l BAQ^m"
.-. AC^ AD = AB^ 10 Iha.
Fjo. 29.
3. Let a be the angle.
Then, since B^'^F^ + Q^ + 2PQ . cosa
92=9« + 6» + 2.9.6.coso,
or 81 =81 +36 + 108 cosa ;
36
1
4. (1) Let AB represent the given force, and let CD, -Si'' repre-
sent the other forces.
D
Pio. 30.
With centre A, and distance CD, describe a circle.
With centre B, and distance EF, describe a circle.
Let M be a point where the circles intersect.
Join AM, MBy and complete the parallelogram AMBN,
Then AM, AN represent the component forces.
(2) Let a and fi be the angles that the
component forces are to make with the
given force represented by AB.
Draw AC making z BAC=a,
and BD making z DBA =^,
and let ^ be the intersection of AC, BD.
Complete the parallelogram AMBN.
Then AN. AM represent the component
Fio. 81. forces.
KEY TO ELEMENTARY STATICS. 15
5. Complete the parallelogram ACDB.
is the middle point of AD and BC,
AC= 13, OC-5, and ^OC is a right angle,
because BAG is an isosceles triangle.
Then ^a«=13«- 52-144 ; A
.-. ^0=12, and /. ^D=24. pio. 82.
6. First, C lies on the circumference of the
circle described with A as centre and AB
as radius.
Next, D lies on the circumference of the
circle described with B as centre and BA
as radius.
7. If P and Q be the component forces and a the angle between
their directions, and if Q be reversed, the angle that it makes with F
is 180** - a. Hence, if B and iSf be the resultants,
JJ2=P« + ^ + 2PC.cosa,
;S[2=P2 + ^ + 2P^ . cos (180' - a)
= pa + C2-2PC.cosa;
.\B^ + 8^^2(P^ + (y), which is independent of a.
8. Let T be the tension of each string in the first case, t the tension
of each string in the second case, W the weight of the picture.
Then TF»= T* + T« + 2T« . cos60%
and TF2= ^^ t^ + 2t^ . C08l20' ;
.-. 2T2 + 2T».J=2e2 + 2«^ x(-i),
or,3T*=e«;
.-. T«:^-:l :3;
.-. r :t =1 : y/3.
i6
KEY TO ELEMENTARY STATICS,
9. Let AQ represent the force of 9 lbs. Then take the formula for
finding the cosine of one of the angles of a triangle in terms of the
sides (Trigonometry, Art. 179),
Fia. 34.
cos.4 = — ^n^ , and we have
2oc '
C082)/1C=
144 + 81-
2xl2x
-36
9 "
225 - 36
" 216 "
189
"216"
7
"8
cos5^D=
144 + 36-
2xl2x
-81
6 "
180 - 81
144
99
"144"
11
"16
10. Let Pi and Pa be any two positions of P, and let be the point
y\ in the arc AGB through which the resultant
passes when P is in the position Pi. Join
CP2.
Then i AP^B^ l AP^, and
/B I APiC^:' L AP^C, (Euclid, III. xxi.)
Now since the two forces are the same at
Pi as at Ps, and act at the same angle, the
resultant must make with them the same
angle in both positions.
.'. PjC must be the direction of the resultant for the position Po.
11. Let the string lie on the quadrant POQ, and let be the centre
of the circle, and OC a vertical line.
Then the forces acting on P are its weight
P, R the pressure of the semicircle acting
along the normal OP, and T the tension of
the string, which acts as a tangent at P.
Let z COP^a.
Then T : P=sino : sin90'. (Art. xxxix.)
.-. T=P.sina.
SimilarlyT=C . sinGOQ=Q, cosa.
Q
.•, P. 8ino=^. cosa, and .•. tana = p-
FiaS
KEY TO ELEMENTARY STATICS.
Hence the eight forces clearly form a
n equilibrium.
13. From Art. zltiil
iF'=(S+4 cos30" + 4 coaeo7,
+ (6 +■ 4 cob60' + 4 eos30=)'
-(8 + 2^3 + 2)' + (6 + 2 + 2^/3)'
-(10 + 2V3)' + (8 + 2V3)*
= 100 + 40 s/3 + 12 + 64 + 32^3+ 13
=.188 + 72^3-4(47 + 18^3);
.■- je=2v'(4T + 18V3).
Also tand^
6 + 2^3 4+ V3 _ (4+ V3)(5- V3 ) 17+ ^3
'10+2V3"6+V3 (5+ V3)C5- ^3)° 22
14. Let F be die (^tcd point.
Take Jtfand iVthe middle points of AB and
r'D. ThelinejoiningJlfandJrpMBea through
O, the centre.
Resultant tAPA, PB=S . FN.
Resultant of PC, PD~iPM ;
.: ^waltmtotPA,PB.PC,PD
=2(re8ultant of Pif, Pitf)
=2x2.P0.
= 4P0.
16. PM represents joint effect of PO,
OM.
NQ represents joint effect of OQ, NO ;
.: PM, NQ cBpreeant joint effect of PQiNM.
„ „ otAD, AB,
otAC.
.: PM, NQ, CA fonn a Bjatem ia-
equilibrium.
i8
KEY TO ELEMENTARY STATICS.
Examples — IV. (p. 45).
(1) 20 : 30=^0 : 5, or, 100=3050, /. 5C=3J inches.
(2) Here P + ^=28, and .'. P : 28-P=3 : 4 ;
.'. 4P=84-3P, whence P=12, and /. Q=16.
Then 12 : 16=7-^C : JLC, whence ^(7=4 inches.
(3) P = JB - C=(14i-i) lbs. = 14 lbs.
Then 14 : i=PO: 2, and .'. PC=56 inches.
Hence ^P=58 inches =4 feet 10 inches.
(4) 3 : 5= JLO- 12 : AC, and .'. ^C=30 inches=2 feet 6 inches.
(6) 10 : Q=18-6 : 18, and .'. ^=15 lbs.
M
5/bs
>\
Fig. 4L
Examples— V. (p. 52).
N
7 /6s
1. 5:7^CN:CM
= G2V:6-GZV;
.-. 30~5CW=7CW;
.-. 0^=2 J feet, and CAr=3i feet.
M
5 m
TV
Fio.42.
N
7J/6S
2. CJf :aZV=16:6
= 3:1.
M
C
IT
FI0.4S.
3. P: ^CNiCMy
aiidP + ^=99;
.-. P:99-P=7:4,
.-. 4P=693-7P;
.-. P=63 lbs., and .*. ^=36 lbs.
KEY TO ELEMENTARY STATICS.
^9
4. C=(16-,12) lbs. -4 lbs. ;
.-. 12:4-»OiV:lfoot;
.•.CiV^= 3 feet.
M
;2f6s
C
Fia. 44.
5. Ci^=36-CAr;
.-. 5:7=36-OAf:(7M;
.-. 5CAf=252-7CM;
.'. 03fs21 inches.
M
C
IT
IG. 45.
6. CM : CN=- 5 : 2, the condition
that one of the arms is 5 inches
longer than the other having no
^ect on their rdaJtive value.
M
2 /6s
M
Fia. 4«.
7. Let P be the smaller weight.
Let X be added to P and taken
from Q, and let P+x act at Ny and
Q-xat M,
Then«Jl^«^.
Hence — - ^— •
Hence -^_^^^,
or,P« + Pa;=^-^
or (P+C)a;=^-P', and .-. a:-C--P.
If we take P the larger weight, x=P- Q.
vv
Fia 47.
N
J/65
N
^
20
KEY TO ELEMENTARY STATICS.
M
M
Fio. 4a.
Fro. 49.
N
8. C7iV=33-a3f;
.-. 03f :33-C3f=3:8;
.-. 80M=99-3CM;
/. (7^=9 inches.
9. CM:8-CM=3:1;
/. CAr=24-3CM;
.-. CM"=6 feet
M
vv
10. Let C be the first position of the fulcronL
Then03f:12-CM=6:3;
D G B N .-.30211=72-6031;
.-. CM'=8 feet.
Now move the weights of 3 lbs.
and 6 lbs. to ^ and B, and let D
A( be the new position of the fulcrum.
6lbs ThenAB=8feet,andl>5=8--4A
Pro. 60. and ^D : 8 - ^ D = 6 : 3 ;
.-. 3^D=48 - e^JD, or u41>=6i feet.
Now ^0=6 feet, /. 2>C=| feet=8 inches.
M
C
7^
N 11. C!af=8in., a2V=38in.
22 lbs
.-.P: 22=38:8;
/. 8P=836 ;
.-. P-1044 lbs.
Fio. 51.
KEY TO ELEMENTARY STATICS.
21
12. Let X be the length of ^C in inches.
ThenCZV:OM=10:15,
%
or, ^:x- 6=2:3; a
3a;
••. 2-=2x-12,
or, OSS 24 inches » 2 feet.
lOlbs
M
N
Fio. 51
C
71
13. C2V:C3f=14:25;
.-. 0^:36=14:26;
.-. 26(W=604 ;
M
^ ^/4lb8
A 25l6s
N
Fia6&
7\
14. Take N the middle point of AB^ this point being equidistant
from and D : a weight of 24 lbs. acting at N will produce the
same effect as that which is a x
produced by the two weights
acting at C and D, and it
may therefore be substituted
for them.
Then, if x be the force acting
upwards at Ay
X : 24=^JV : AB
= 1:2;
.*. 2x=24,.'. x=121bs.
N
Fia54
B
TV
'12ll>S
22
KEY TO ELEMENTARY STATICS.
16. (1) Draw ON at right angles to BQ,
ThenO^:OJB=V3:2;
Fia. 55.
.-. ON-
5^3
2
And P:Q=OiV:0^
or4:Q=.^:3;
8^3
lbs.
Then since triangles AOMy BON are
r similar,
(2) Draw OM, OiV at right angles to AP, BQ.
A ,
ONiOM^OBiOA
=5:3.
p X NowP:C=OiV:OM;
Fia 58.
/. 3 : C=5 : 3, and .-. Q=\\ lbs.
(3) Draw OM, ON at right angles to AP, BQ.
AO 3
Fio. 57.
Then 0M»
V2 ;72
••^•'^-2 V2' V2 2 '
- 2x3x2 2 X 3 X 2 X V2 ^g ii^
•■• '^"" 6, X V2 " 6l^T " 6
KEY TO ELEMENTARY STATICS.
EXAMPLB— VI. (p. 661.
Then if C be the centre of gnvit^
2. Tha weight of 4 lt». at j1 and % lb«. at ^, the middle point t£
AB, are equiTalent to a w^ght of 6 Ibe.
at D, AD being one-third of Alf, t
Sinchea.
.'. DB=10 inchea, and tha centre of
gravity of 6 lbs. at i> and 1 lb. at B ia
f of 10 inches, or 1^ inches from D.
Hence the centre of gisTit; of the i\hs
system IB 3^ inches from A.
3. By Art. 71, if ^ be the centre of graTity, and D be taken
the fixed point whose distance from . - f. (.
Ifwe want to find in in<^iee. '~~
„^ P.AD*F.BD*P.CD
4. To find the distance of If the centie of
gtavi^ from A in feet.
.„ S.ABi-iA0+2AD+l.AE
^^~ 16+8+4+2+1
8 + 8 + 6 + 4 26, .
mn
S 8 4 £ 1
24 KEY TO ELEMENTAR V STATICS,
5. The centre of gravity of B and C lies in
BCy and it also lies in the line AD passing
through the centre of gravity of the triangle.
Hence it lies at D, the middle point of EC.
.\ B^Cy and similarly it may be shown that
*^A^B^a
6. Let BG be the base, ABC^ DBC any two of the triangles.
Draw A My DM to M the middle point of BC.
Let be the centre of gravity of ABC,
and draw OP parallel to AD, cutting DM
Then shall iV be the centre of gravity of
'b \ /mX \ P DBG.
For since ON is parallel to AD,
DN:NM=AO: OM (Euclid, VI. ii.)
= 2:1.
7. DEy joining the middle points of AB, ACyia parallel to BC.
Hence AF bisects DE in 0.
.*. the centre of gravity of DFE lies in FO,
that is, in AF, which passes through the
centre of gravity of ABG, Also, the distance
of the centre of gravity of DFE from F is
f . FO=i . AFy which is the distance of the
K* centre of gravity of ABG from F. Thus the
triangles have the same centre of gravity.
Fig. 64.
8. Taking the diagram of Example 6, suppose and ^ to be the
centres of gravity of two of the triangles.
Then since A0^20My and DN-=^%NMy
.\AO:OM=DN:NM;
.*. AD is parallel to ON,
KE Y TO ELEMENTAR V STATICS.
25
9. Let P, Q be two equal particles ; P', ^, the same partides in
new positions.
Now, since FF'^QQf, and the angles of
the triangles FF'Oy QQ^O are equal, each
to each, / y^o
.-. PO^QOy and rO^Q'O ;
. *. is the centre of gravity of the particles D Q. Q' c
in both positions. F,o. ^
10. Let D be the centre of gravity of A and By and E that of A
and C. Let CD, BE intersect in 0. A
Then the centre of gravity of A, B, C lies
in CD and also in BE,
.'. is the centre of gravity of Ay B, C.
Hence JY, the point in BC where AO
meets BCy must be the centre of gravity of
B and C B N C
Fio. 66.
11. Let be the centre of gravity of the equi-
lateral triangle ABC, Then AD bisects BC at
right angles.
Hence OB^OC;
and similarly it may be shown that OA = OC, ^
.'. is the centre of the circle.
Pio. 67.
12. Taking the diagram of Example 11,
J5Z)= CDy because is the centre of gravity of the triangle,
and 0B= OC, because is the centre of the circle.
.-. A ODB= L ODC^a, right angle.
.*. AB=ACy and similarly it may be shown that AB—BC.
,\ ABC is equilateral.
13. To keep the line, passing vertically through the common centre
of gravity of his own body and the weight, within the base covered
by his feet.
26
KEY TO ELEMENTARY STATICS,
D
"7^
B
3/6s
Fio. 68.
%[hi
14. Let AB be the rod, its
middle point, D the edge of the table.
Then we have 3 lbs. acting at O,
and 4 lbs. at B,
Hence, since OB =7 inches,
0Z)= 4 inches ;
and .*. ^i>=ll inches.
15. Let the weight of each particle
be P.
Then P at -4 and P at £ are equiva-
lent to 2P at M, the middle point of AB.
Also, P at and P at i> are equivalent
to 2P at 0, the middle point at CD.
.'. the centre of gravity lies midway be-
tween and M.
16. Let A be the point of suspension.
Then the vertical line AD passes through the
middle point of BG, and since BC is horizontal,
the angles at D are right angles.
Hence AB=AC. (Euclid, I. rv.)
17. Let 0, P be the centres of gravity, A the middle point of th«
common side, C the centre of gravity of the two.
OA will be perpendicular to the side of the square.
Draw BD at right angles to the same side.
Let a be the side of the square, b the altitude of the triangle.
ah
Fio. 69.
Then area of triangle »»•
Area of square s^a^
.\^:a^=OG:BC
~ OA : BD (by similar triangles)
a
Fio. 7L
"2 '• 3 '
KE V TO ELEMENTAR V STA TICS,
27
18. ABC, A'BfC are the two positions of the triangle, CDBU
being a vertical line bisecting AB and CA',
Let iDCA=e, AC^Xy AB^y,
Then z CAD^By and z ^DJ5'=2d,
/. z ^'J5'J[>'= z ilFi>=»90'- 2d,
and z ^^'2/=- z 5ilO=d ;
.-. z^'J[>'B' = 180°-(90*-2^-d=90* + d.
'^^^ 8in(90° - 2d)~A'iy~ x '
8in(90° - 26)
co6$ _2y
or.
1/
cos2d
X
or
apy ^2y.
-1
^2y
'2x2-y« a; '
•. fl5*=4a;'-2y';
.-. 2y«=3»« ;
/. x:y=V2 :V3;
/. JC:CB«V2:1.
Fio. 72.
19. Place 3 lbs. at A, 4 lbs. at By and 5 lbs. at C.
These are equivalent to 3 lbs. at Ay By C, together with 1 lb. at B,
and 2 lbs. at C. /^
These are equivalent to 9 lbs. at 0, the
centre of gravity of the triangle, together with
3 lbs. at Dy one-third of the distance of C
froni^.
Hence the centre of gravity is one-fourth of
the distance of D from 0, measured from 0. B DC
20. Place 2P at B, 2P at 0, and P at A.
These are equivalent to 4P at Dy the middle
point of BCy and F at A,
Hence the centre of gravity lies in AD at a
distance from D one-fifth of AD,
28
KEY TO ELEMENTARY STATICS.
21. Let A be the point of suspension.
The centre of gravity of the parallelogram
being the point of intersection of the diago-
pnals, JC is a vertical line, and since the
diagonals of a rhombus intersect at right
angles ;
.*. BD is honzontaL
22. Draw MON^ BOS through 0, the point of intersection of the
diagonals, parallel to the sides of the original parallelogram.
Then MONy BOS wiU bisect the sides.
Hence the centres of gravity, A, J5, C, D,
of the four triangles into which the paral-
lelogram is divided by its diagonals (all
of which triangles are equal), are equi-
distant from 0. Now the lines joining
the extremities of the straight lines AC,
BT>f which bisect each other, form a parallelogram, as can be easily
shown.
Pio. 77
23. Let ADCE be a vertical line.
'i MNC= L NAC+ACN
^iCAD+ lA'CE.
Now since AD bisects BCi
.-. DC^^BC^AC;
And since CE bisects A'B^, E is the
centre of the circle described round
A A'BfC;
.-. ECr=-EA'.
And .-. z A'CE^' l at ^'= z A ;
.-. zMyC?'=46'+A
KE V TO ELEMENT A R Y STATICS.
29
24 Let C be the right angle, CD the vertical line, bisecting AB
inD. C
Then z CDB^ i DCA+ i CAD
= 2 z DAC (for DA=DB^DC).
But z DAC+ L CBA^W ;
or, zD^O+6zD^C=90%
and.-. zi>-4 0=16°;
.-. zCDB=30^ A
Pio. 78.
25. First remark that under the conditions of the question, if AB^
AC be the equal sides of the
triangle, and the centre of
gravity, z OBC^ z 0CB=46",
since OD=BD=DC,
Next, let BMCN be the
vertical line passing through
the points of suspension, and
let B^C produced meet BC, or
BC produced, in JR.
Then ^
180°- z CJB(7= z CBM+ z RCB
= z C5M+ z ^(TJ^
=:46• + 46•=90^ Fio.7d.
26. Since AC=2BC, C is the centre A
of gravity of the weights acting at A
and B,
,\ C m the centre of gravity of the y
three weights.
ilb
3dts Zlbs
Fio. 81.
27. Let P be the weight of each of the particles acting at A, B, D.
Then P at B and P at D are A
equivalent to 2P at 0.
Then if G^ be the centre of gravity
ofPat^,and2Pat 0,
AG='^ of AC=^^ o{ AD.
B G
Fia 81
i
30
KEY TO ELEMENTARY STATICS.
P 28. The limit in length of AB is attained
when the diagonal ^C is perpendicular to
BC, for then the centre of gravity lies just
over C.
In this case, since z ^1BC=60*,
^-B= 250= 12 inches.
29. BE bisects OC in K
Then the forces, each equal P, acting at
0, Ay By Cy DyEaxe equivalent to 2P at O,
2P at Ny P at 0, P at 0, and therefore to 2P
at 0, and 4P at N.
Hence if G^ be the centre of gravity, it lies
in OGy and is such that
0(?=|- of Oiyr=y of OC.
30. The centre of gravity of the body cpincides with the centre
of gravity of the two parts taken together, and as this must be
in the line joining Gi and G^, it follows that G must be in that line.
31. is the centre of gravity of the square, G of the triangle, H of
the figure AECD. Let 2a be the side of the square, and let EB=^x.
0, ,C Then 4a^ - ox ; ax=OG : HO
^MNiEM
2 X
=«-3-T^^"^'
.-. 4ahi-aoi^-4a^ + ah:=ahi- -j-, and hence
x=3a±a^3.
Thus5^=aV3 ( V3- l),and ^JS?=a ( V3- 1),
md.\BE:AE= ^3: 1.
32. Let Af, i\r be the centres of gravity
D of the equal triangles ABD, BCD.
Then since equal weights act at M and
Ny we may find the centre of gravity of
the trapezium by taking the middle point
Fio. 8$. of MK
KE V TO ELEMENTAR V STATICS.
31
33. Let be the centre of gravity of the square, N the centre of
gravity of the triangle, 2a the length of a side of the square, x the
distance of A from 0. *
Then area of square =4a',
area of triangle ^^a {a + x),
and AO: ON==a (a + sc) :4a*-a (a + x) ;
or,a:a-^ -=a' + <w; :Sa^~ax;
or, 3fl5 : 2a — a;=a + a; : 3a~» ;
or, 9aac - 3a^=2a^ + ax - sc*.
/. «* - 4aa;= - a', v p,o. 87.
and a;=2a±V3a.
34. Let J-B be the side on which the triangle will not stand, the
centre of gravity of the triangle, D the middle point of AG,
Now z ABC is an obtuse angle,
.'. ACia the greatest side.
And '.• z CDB is an obtuse angle ; ^-
.-. CB is greater than CD ; ^^^
/, CB is greater than AB.
Also, •/ DBA is an obtuse angle ; ^
.*. AD is greater than AB,
,\ AB is less than half of AC
35. Draw DE^ joining the middle points of AB, AC,
Then triangle ADE is one-fourth of triangle ABC,
Let jR be the centre of gravity required, the centre of gravity of
triangle ADE, G the centre of gravity of triangle ABC.
Then -^ X 0(?=-| X 22Gf ; ^
,\ Oa^ZBG,
,\Aa-'AO^ZAB-'ZAa]
.', 3^iJ=^^, or, AR=^ oiAF,
36. D, the middle point of BC, is equi-
distant from Ay By (7, if BAC be a right
angle.
.*., if be the centre of gravity,
A0=^^ of AD'' J oiBD^% incbes.
32
KEY TO ELEMENTARY STATICS,
37. Let BQC be the equilateral triangle. The limit of equilibrium
is reached when the centre of gravity of the two figures lies in the
vertical line BG,
B
M
N 0.,^,..^-'^^
Fio. 91.
Let 2a be the side of the
square, x the greatest height of
the triangle, Af, the centres
^ of gravity of the two figures, N
the centre of gravity of the two
together.
Then 4a« . MN-^ax. NO.
X
/. 4aS=aa; x --, or, 12a2=:a;«, and .-. x^2^3,a.
38. Let 0, E be the centres of gravity of the figures, G the com-
mon centre of gravity.
Then 4a^. OG^ ^3.a\GE;
or, 4 . 06?= V3 . {OE- OG) ;
■"^ > or(4+ V3.)0(?= V3.(a + -i-.V3.a)
=a(V3 + l):
Fia 92.
39. The triangle ABE is one-third of the quadrilateral DBCE.
Let ^ be the centre of gravity of DBCE^ M that of ADE, that
of ABC.
.'. OM=Z.ON;
or, AO-AM'^S. ON;
or, jAF'-^AF^SON;
.'. ON^^AF.
.: NF=^AF- ^AF= ^.AF;
.-.NF^^.HF.
£^£y TO ELEMENTARY STATICS.
33
4
37
4
fl
40. (1) The given Bystem is eqniTalent to the system npresentoi
in this diagram, if we combine eqmil parts of the
forces, acting at the centrea of tiie outside squares
on the light and left, in the centre of each middle
And this system is equivident to the system repre-
sented in this second diagtam, if we combine the forees
Tertically.
.'. the centre of grant; is the middle point of the
middle square ; that is, the point of intersection of \
the diagonals.
(S) The new system is equivalent to the ajstem represented in
fignre 96, if we combine the forces horizontally.
And this is equivalent to the ^stem represented in '
figure 97.
Hence, if we take the struct line joining the
centrea of the sqnaies marked 9 and I in the origins]
diagram (since we have forces in the ratio of 3 : 1 act-
ing at the middle point and the lowest point of this
line), the centre of gravity of the Ejstem will divide
the line joining the centres of 9 and 1 in the ratio
5:4.
41- Combining horizontally, the system is equivalent
9i at if, 6^ at 0,
12 at £^, 5 at j;
&i at D, 3| at £.
Hence if R and Q be the position of the points
, -where tiiese forces, taken vertically, may be col-
lected, a. b^ng a side of the sqoare.
z
4.
It
*
2
13 1
0.
^
=
^«
fl2
HB—
3 33a 11a
H M
R
(4
— —
1
3
34
KEY TO ELEMENTARY STATICS.
Also, since 0, the centre of gravity of 30 lbs. at R and 15 lbs. at
Q, is at the distance from i2 of | i^Q,
ON^Ua^^Ha^^a,
n-KT nrk , ^ / tt-d />in\ 41a 2 /11a 41a\ 14a
42. Let a be a side of the large square, N the centre of gravity of
the large square, It the centre of gravity of the portion removed, M
the centre of gravity of the remainder.
■^ Then MAT x(a«-^)=i2JVx~^;
.'. MJVx 6a2=jBJVx 4a«.
y^
/
Fio. 99.
.-. MN^^MC, and .'. CM =i^, of CG,
15 ' 30
43. Let F and B be the middle points of J-B, BC ; D the centre of
gravity of the straight bar ; G the centre of gravity of the bent bar.
Then FB^4 feet, and BB=Z feet, and .*. FE^h feet.
Then J^(?=y of FE = y ^®®*> ^^ -F!D=3 feet.
a2 + 62-c*
Take the formula cosC=-
2a&
D B
Then coaDFG=
(Trigonometry, Art 179.)
FD^+FG^-Giy^
2FD . FG '
225
or, -=- =
9 + ^-GD*
Pio. 100.
_4
5
or, — =
whence GD=
2x3xy
441+225--49gD2
2x3x15x7 '
9V2
7
KEY TO ELEMENTARY STATICS,
35
,C
44. 0, the centre of grayity of the complete hexagon, is equidistant
from the angular points. Let the side omitted be indicated by
the dotted line, and let the distance between
M the middle point of this line and the opposite
side be a.
Then if G^ be the centre of gravity of the
mutilated figure,
OOx5=03fxl;
"^^ 6 10
45. The centre of gravity of a triangle coincides with the centre of
gravity of three equal weights placed at the angular points. Hence the
weight sustained by each prop is one-third of the weight of the triangle.
46. Draw a radius CD from any point, and place the weight of 5 lbs.
M
Fio. 101.
at a Produce CD to E, so that ^2>=f . CD.
Draw the chord AEB at right angles to CE,
and place the weights of 3 lbs. each at A
and B, These are equivalent to 6 lbs. aoting
at E, and 6 lbs. at Ewill balance 5 lbs. at CE,
because DE : 0D=5 : 6. p,o. 102.
47. Let the line drawn parallel to the base cut the line ADy joining
the vertex of ABC to i>, the middle point of BC, into two parts,
m and n, of which m is the part nearer to the vertex.
Then area ofAEF : area of ABC =m^ : (m + n)^
Euclid, VI. xix.
Let and G be the centres of gravity of the
two triangles, and B the centre of gravity of
the quadrilateral EBCF,
Then GBx{{m+ny-m^^OGxm^.
i jn 2(m + n) j ,^ „> 2 „
^ o } 6 Pio. 103.
. -P_2 { (TO+»)»-m»}
^
36
KEY TO ELEMENTARY STATICS.
48. Let A be the point of suspension. Then,
by the conditions of the question, AD bisects BC
at right angles. Hence AB=^AC (Euclu), L iv.) ;
and .*. the four sides are all equaL
49. Place w at Ay 2w at B, 3io at C w at
A and 2w at B are equivalent to Zw at JD,
AD being equal to twice DB.
Join DC, and bisect it at G, then G is the
centre of gravity required.
AhoAG:GE=6:l
CG:GD=^l:l
BG : GF=-2 : 1.
60. Let G be the centre of gravity of the Utnin^ ^
and D the middle point of AE the perpendicular on
BC. Let X be the weight placed at A^
ThenxxAD=^6xDG;
AE ^ [AE AE\
or,xx — =6x(^— -_j;
or
6^
6
^|.=Z^and.-. a;=21bs.
Fio. lor.
51. Let A be the point of suspension. Then
AD is perpendicular to BC Let P and Q be the
weights.
Q Then P:Q=aD:5i>.
lSowBD:BA^BA:BC} _ ^
and CD : C^ = C^ : 50 1 ^^^°' ^- ^^•
.*. 5D= jg^ and CD= -g^ ;
/.FiQ^CA^iBA^.
KEY TO ELEMENTARY STATICS.
63. At the limit of equilibrium AC is vertic
.-. the iucliniktioo of the ptane is 45°.
If the incUnation of the plane be greater
than 45°, the centre of grayity of the block
will be outside the base, and the block will
63. ^ is the centre of gravity of the two
smaller squares, and H is the cenire of gravity
of the sqnaie on the hypotenuse.
Now the weight of the two squares collected
at ^ is equal to the weight of the larger square
collected at D, and therefore the common
centre of gravity is at 0, which bisects AH,
and also bisects the hypotenuse of the triangle.
Examples— VII. (p. 87.)
1. Let AT>—x feet.
Then I>£-(S-3:) feet,
andic-e (S-i), .-. a:=3.
Pressure on the fulcrum =4 ]hs + 6 lbs = 10 lbs. 4-
2. DB:AJ)=1:^. I
P+Q=361b3.
NowP:g = DB:JD;
.■. P:36-P=7:9, .-. 9P=352-7P;
.-. P=15ilbs.,aode=20ilhs. P p^j^,
3. Let X be the weight of the rod, which may be supposed b
at M, the middle point of the rod, ' ' "
Thenx:3-DJ7:Mi>
=4:3J;
.•.ia!=12,or,a;=V=3?Ilw.
38
KE Y TO ELEMENTAR Y ST A TICS.
A6
A D
B
Fio. 118.
4. Let AD^x inches.
Then 6^D + 6CD=10DB,
or, 6a; + 6 (10-a;)»10 (7-a;),
or, 6a; + 60-6a;=70-10a;;
.'. xsl inch.
Pressure on fulcnim = (10 + 6 - 6) lbs.
» 10 lbs.
so
w
Fig. 114.
5. Regard B, where the rod rests on the shoulder of one man, as a
fulcrum, and let x be the pressure on the
A C B shoulder of the other man ; w the weight
^ of the rod, acting at its centre of gravity,
distant from B one-third of the length of
the rod.
Thenx:ir=^C:^^
= 1:3
.'. 05= one-third of w.
Hence pressure on ^ = two-thirds of w.
6. Let AB be the original length of the lever, D the fulcrum,
DB = one-third of AB.
Q_ Then C=2P.
Now let Q act vertically upwards
at By and let C be the new position
4? of the fulcrum.
Then WC=AC
B
Tia. 115.
^AB+BC;
.-. BC=AB=^3DB.
vv
\
7. 3f is the middle point of the lever.
A D_M Let ^2>=x feet.
Thenl0x=50(3-a;),
10x=150-50r,
and.'. 2=2J^feet.
Also pressure on the fulcrum =60 lbs.
10
60
VI0.11&
KEY TO ELEMENTARY STATICS,
39
8. Collect 1/7, the weight of a
the beam at the point N round
which it balances.
Then w x DiV=100 x DB,
or, ii;x8 = 100x2;
/. IT = 25 lbs.
Ji
11/
Fio. 117.
D
B
JOO
9. M is the middle point of the rod, 2t
w is the weight of the rod.
Then 2 J x ^D = w x 3fD + 2| x BZ>,
or, 2J X 4=ir x 2 + 2| x 1 J,
or, 10=s2w + 4, and .*. w=^ lbs.
Also,
pressure on D= (3 + 2§ - 2J) lbs. =3J lbs.
M B
w 24
Fig. 118.
D
TV
10. Regard the peg D as a ful-
crum, and let x be the pressure on
the peg C.
Then SxAD=x>iCD + 6xBD, A
or, 3x5=4« + 5xl ;
.'. a=»2J lbs.
Hence
pressure on D= (8 - 2^) lbs. = 5 J lbs.
Next, let 2p be the upward
pressure applied at N, a point
2 feet from B,
X
AT
Then pressure on each peg
=i(8-2|))=4-p.
Make D the fulcrum, as before. ^ [
7\
Fig. 119.
A.
B
N
Fio. 120.
Then 3 x AD={^-p) x DC+2p x ^2) + 5 x DB.
15=16-4p + 2p + 5, .-. 2;>=61bs.
B
KEY TO ELEMENTARY STATICS.
11. Let DB-x iocbet.
or,2(17-a;) + 7(15-i) + B(13-jr)-
or, 34-2x4- 105-71 + 65 -Si-ai;
.-.a; -12.
12. The adrantoge Till be increaaed, becatue the dititauce of the
power from the fulcrum b increaaed.
13. Let i£ be the middle point of the bar, and suppose the pressure
on the prop it to be twice as great oa that on A, the prop that is
1 foot from one end of the bar.
^ Then pressure oaA— -»-•
BegBjd B aa the fulcrum, and
W
make an upward force of -^ acting
•if] at A repreaeut the reliance of A.
kAB'W^MB
- W. [AB-Am = f^- (JB-4) ;
.-. AB~ZAB- 12, or, AB-Q feet.
14.LetCbetliepointof
application, CB = x inches.
Then
KEY TO ELEMENTARY STATICS.
41
15.xxOD=8x^D + 8x^J>,
or, 16a; =8 (18 + 12);
/. 2a; =30, or, x=16 lbs
The force, being less than
the sum of those which it
balances, acts at a mechanical
advantage.
B
1
TV
t
Fig. 124.
16. Make B the fulcrum, and re- l
gard the resistance of A us an up-
ward force »a; stone. M is the
A
M
A
3
A
i i
Fio. 125.
middle point of the beam.
Then x x ^5=4 x 2)5+ 16 x MB,
6a;=4x4 + 16x3;
.*. a;=10f stone;
and .*. pressure on B=(20- 10§) stone =9 J stone.
17. Suppose the weight supported by the man at B to be p, and
the weight supported by the man at ^i to be rp.
Then the weight of the beam acting at the middle point Jf is (r + \)p.
Regard B as the fulcrum.
Then rp x ^5= (r + 1);> x MB,
Let 2x be the length of the beam.
Then rp. (2x- a- 6) = (r +!);>.(«- 6),
or, r (2x— a-6)=ra; + x-6r— 6,
or, 2ra; - ra=ra; + a; - 6,
or (r- l)x=»ra- 6, and .'. 2a;= -^^^— <*
^ ' r-1
Ay?
M
A^
^B
V+flp.
Fio. 126.
18. Let M, .N* be the middle points of the axes of the cylinders, and
the point where the axes meet. M Q |s|
Now since If = f ft., and NO = « ft,S
and since 15 x |=9 x |,
it follows the cylinders balance at
0, the point of junction.
S
3)
iS
"E\<i.vn.
42
KEY TO ELEMENTARY STATICS.
15
BD
130
Pio. 128.
25
19. Let OZ>=a; feet.
Then 15 X (12 -a) + 130 (6-a;)=25a;;
or, 180- 16a; + 780- 130a; = 26a:;
or, 960 = 170a;;
20. Let D be the point at which Wy a weight equal to the weight
of the rod, is placed.
Let M be the middle point of the rod.
f
M
T
tlT
T
Fio. 129.
4w
Then pressure on^ =§ ot{w + «<;)=-
o
Regard B as the fulcrum, and repre-
■J^ sent the length of -4 B by x.
4W7 / ^ TXN X
Then -5-xa;=i/7x (a;--4X>)+tr.— ;
o Z
4a; ^ T^ . «
a;
.*. ^D =—-—=—= one-sixth of the length of the rod.
2 3 6 ^
X
Fio. 130.
152
B
21. Regard B as the fulcrum,
and let a;, equal to the weight
sustained by A, act upwards at
A.
Then x x J[j5= 152 x 2>j5,
or XX 38=152x14;
.*. x=56 lbs. ;
and .'. pressure at JB=96 lbs.
22. Let w be the weight of the beam, which may be supposed to
A C M D act at Z>, roulld which the beam
balanced in its first position.
Then 200 x AM+ 20 x CM^^w x MD ;
or, 200xl2 + 20x8=2ti;;
Fio. 131. .'. 2t(;=2560, or, w=1280 lbs.
200 so
Y
ur
KEY TO ELEMENTARY STATICS.
43
23. Let M be the middle point, D the fdlcmm at a distance of x
feet from the end N,
N-
A B
•f y >r
C E p M
3 5 7^ iSO
Fio. 182.
Then 3^D + bBD + 7CD+ 9ED = 150MD.
Or, 3(x-2) + 5(x-4) + 7(x-6) + 9(a;-8) = 150(10-a:) ;
.-. 174x«1640 ; or, a;=^9|f feet
24. (1) Let If be the middle point of the lever, W the weight of
the lever, 2> the fixed point, B the end where P, when suspended
from it, balances the lever.
Then WxMD=PxDB,
Let A be the point where P
is placed to balance nP acting
at B.
Then
PxAD+WxMD^nPxDB;
OTy PxAD-k-PxDB=nPxDB;
OTfAD + DB=nDB;
.'. AD=(n-l)DB,
(2) Here^D=10xi)j5;
.'. 10=n-l ; or, n«»ll.
M
D
W
Fio. 13S.
nP
25. Let If be the middle point
weight mnst be attached to somepoint
in ^D,and that it may be as small as
poesible,^ it most be made to act at A,
Let X be the additional weight
Then
(x + 3) X ^2>=:2 X DAf 4-4 X DB ;
or, x + 3 = 2+12, and.'. x=5ll lbs.
It is clear that the additional
M
X-hS
2
Fio. 134.
44
KEY TO ELEMENTARY STATICS.
26. Let W and 3 TT be the weight of the rods, JVand M the middle
points of the rods, x the weight attached to W, D the fulcrum, A
and B the points in the horizontal
line through the fulcrum, through
which the vertical lines through y
and M pass.
Then since z ADN = i DBN,
and z DAN= l DBM ;
. *. triangles ADN, BDM are similar.
Then Pr+a;;3Tr=2)5 ^D
=3.1;
W+x^9W;
/.x=SW.
FiO. 185.
27. Let M be the middle point of the rod, (7 the first position of
the fulcrum, D the second position of the fulcrum, W the weighty
and 2x the length of the rod.
5*
C D
M
B
T
Fio. 136.
Then5f x(2a;-2) = Tr(a;-2),
and6(2x-3) = (Tr+l)(x-3)
Hence56x-66=5Trx-10Tr, )
lOx-lb^Wx-SW-^-x-d, )
OT,bex-b6 = bWx-lOW, )
bOx-7b=-bWx-lbW+6x-l6;)
.-. 6x + 19 = 5Tr~6x+16,
or, 5Tr=lla;+4.
Substitute this value of STT in (3), and we get
56a;-56=llx2_i8x-8,
and from this we find a =6, and .'. 2x=12, and W=14,
(1)
(2)
(3)
KEY TO ELEMENTARY STATICS,
45
Then Tr(a;-2)=5x2 )
Tr(a;-l)=llxl )
Wx-W ^\\S
7V7^
28. Let TT be the weight of ^ [^ q
the rod, 2x the length of the
rod, C and D the two positions
of the fulcrum.
M
w
Fro. 18T.
W^=l, and.*. 2x=24.
M
TV
29. Let Jlf be the middle point
of^i;.
Thenl2x AD=8xDB+xx dm/'-
or, 12xll«8x4 + a:x5;
.•. a;=28f lbs.
f2 Fio 188.
30. Let DB^x.
Then9x7 + 7x6 + 6x3«3x2 + 6x4 + 7x8+x,
D B
Fro 180.
or, 63 + 42 + 15 = 6 + 20 + 56 + «;
.'. a;«38 feet.
31. From D the fulcrum draw DB at right
angles to the direction of x, the force acting at
the longer arm.
Then since z CDB-60*,
/. CD-2DB.
Kowa;xi>£«6x^
or, a; X j >-6 X 3, and .'.a^— 7J lbs.
^/
8
46
KEY TO ELEMENTARY STATICS,
Fiu. 141.
32. From D the fulcrum draw DiV at
right angles to the force of 10 lbs.
Then AB : DiV= 2 : ^3,
and 10xDiV=20x4;
.-. DiV^=8, and /. -4D= ^ = —3-^
.'. the length of lever is — ^ feet
33. One way clearly is to reverse the position of the rods, making
the 1 lb. weight of the one correspond to the 9 lbs. weight of the other.
» Q Another way is found in the following
manner : —
Let D be the fulcrum.
Let X == the distance of the end of the
\ t
\j 1/ f Y ^' upper rod from the fulcrum.
' ^ ^ Then
Fio. 142.
lxa; + 3(a;-l) + 6(a;-2)x7(a;-3) + lx24-3x 1
= 9(4-a:) + 7xl+9x2,
or, x + 3a;-3 + 5a;-10 + 7a;-21+2 + 3=36-9a; + 7 + 18,
or, 16a; -29 = 61 -9a;, and .*. a;=3f feet ;
.". the 1 lb. weight of the upper rod projects 1| feet beyond the 1 lb.
weight of the lower.
I
34. Draw BD at right angles to the direction of the force of 4 lbs.
Then 2 X ^£=4x51)
^4xAB,amBAD;
.-. sinJ5^D= J, and .-. z BAD ^30\
Also, since the pressure on the fulcrum
acts along AB, if i2 be this pressure,
42=222 + 2*, or, 2^2=12,
Fjo. 243. and .'. i2=2 V3 lbs.
KEY TO ELEMENTARY STATICS.
47
A B D
35. Let FT and 2Tr, the weights of the beams, act at M and N
their middle points.
Let P be the weight required.
Let ABDC be a horizontal line.
Then PxAD+Wx BD=-2Wx DC.
Now BD=ND, and /. AD-^DC,
And RM^MD, and .-. BD=\AD,
Hence Px ^D+ WxiAD^2Wx AD,
Fio. 144.
W
3fV
or, P+ 2-=2fr, and .*. P= ^ •
36. Draw MEN At right angles to the directions
of the weights, and BD vertical ^
Then since the weights are equal BM^BN;
.;. AD^DC, (Euclid, VL it. Ex. 1.) M
Hence, since ABC is a right angle ;
,'. DB'^AD^DC. (Euclid, IIL XXXI.) p p
.% z DAB= L ABD, and z DCB^ l DBC, fiq 145,
37. Xet ADB be the lever and D the fulcrum,
CDB a line perpendicular to JIP, PP, Jf the a
middle point of ^iP, MD a vertical line.
Then since AP, MD, BP are parallel ;
•. CD : DE=AM : 3fP.
(Euclid, VL II. Ex. 1.) I
.*. CD=DE, and .'. there is equilibrium.
Fio. 146.
38. Let MAN be a horizontal line. Then since
AB and AC sie equally inclined to the vertical,
they are equally inclined to the horizontal ;
^ .\ lMAB^lNAC',
.*. MBA, NCA are similar triangles.
l!heaP:Q=:^AN:AM
^ACiAB
*1 :2.
irva,\«
48
KEY TO ELEMENTARY STATICS.
39. Let D be the fulcram, MBN a horizontal line.
Then i>lf =
AD
X
Fio. 149.
and DiV=^.D5;
.•.xxi|=16x^xl2;
16x^3x12x^2 16/i/6,v
== 27T8 ^3 ^^'
40. Let P and Q be the forces acting
along AB and AD,
Draw CAf, CN perpendicular to -4P,
Then since z ^PC= z ^2X7 ;
.-. iCBM=iCDN;
.*. triangles BMC, DNC are similar ;
.\F\Q^CN:GM
U
T
f2
Fio. 150.
Examples — ^VIIL (p. 96.)
^ \. W will be greatest when the
1 moveable weight is at D, the extre-
J mity of the steelyard.
' Then ^GD x 12a(?= W. CA,
or, ^x 33 + 12 xl4=5fr,
and .-. fr-42f.
W
2. First, when G^ the centre of gravity is in the longer arm.
Let P be resting at the graduation marked n.
Then the steelyard says that W'^nP, or, W^2n lbs.
BD GCOA BD OCGA
W
P
Q W
^
Fio. 161
But really 2. CD + Q. C(7-Tr. CA (a),
a,ndQ,CG=l.CO;
.'. 2CD+C0'-'W. CA, or, 2 (OI)-CO) + CO>
W.CA.
KE Y TO ELEMENTAR V STATICS,
49
Also, OD^nCA ;
.\2n.CA-C0=^W.CA;
CO
,\ Tr=»2n-7=-j, and /. W is less than 2n*
Next, when lies in CA, the equation (a) becomes
^D'=W.CA + CO,
0T,2 (OD + CO)^W.CA + CO;
.-. 2n.CA + C0= W.CA;
CO
,\ Wss2n+rj-2, and .*. W is greater than 2n.
3. PxCD+ — xOGf-TTx^C.
P
Let acss length of rod.
P3a; , P «. TIT- Jc
4 p 4 4
.-. 3P+^«fr, or, Tr=(^i-^y.
W
P
Fio. 152.
P
4. There is no limit to the increase in the magnitude of P, where-
ever the fdlcrom may be, provided that the graduations can be made
soffidently small ; but there is a limit to the decrease in the magnitude
of P, except when the fulcrum is at the centre of gravity, beoiuse P
may be too small to balance Q when P is made to act at a point in
CA.
5, Let B be the weight suspended at Q to make the instrument
correct for a moveable weight nP,
TheaP.CD-i-Q.CG^W.AC, (1)
AadnP.JDD+(Q+B).Ca^nW.AC, (2)
G IC
T
Fia.lS8L
Henoe, subtracting (1) from (2),
(n-l).P.CD+B.CO^in-l).W.AC,
OT,B.Ca^(n-l)(W.AC'P.CD)
=(n-l).^.0(?;
.\B^{n-l).Q,
D
from(l)
50 KE Y TO ELEMENTAR V ST A TICS,
6. If a be. the real weight,
a; : 38 = 20 : 19, and /. x=40 lbs.
7. The arms of the balance are as 14 : 16 ;
.*. when the body is placed at the end of the longer arm, apparent
weight : 16=16: 14;
/. apparent weight = — rj— =18? ounces.
8. Let X and y be the lengths of the arms, 'p and q the apparent
weights.
Thenjpa;=y, and qy=x;
y X
^ ^ X y'
x^ + y^ 13
••* g — y 9
"^^'^ 6 ^ 144 "144'
13y 5y
or X — • — =s — ^ •
^^'^ 12 12'
••• |"=f or §.
9. He gets 15 ounces for 3s. 9d.
.*. for 16 ounces he pays — ^-— d., or, 48d.,
that is, he buys at the rate of 48. per lb.
10. Take the diagram on page 95 of the Statics,
P= l&n ounces. Let the graduations nm from to ^.
When ir= 1 ounce, ?-^=~» and .'• OC=^:r^^. • AO.
AC IGn 16/1 + 1
When ir=2ounce8, -—=,-—, and.'. OC^r^ r -4 0, and so on.
AC 16n' 16w + 2
KE V TO ELEMENTAR V STA TICS. 5 1
Examples — IX. (p. 99.)
1. Let X be the radios of the axle in inches.
Then3:18=x:3xl2;
3x3x12^.,
.-. x= — r^ =6 inches.
2. Let X be the power in lbs.
Thenx:3=2:6;
.*. 6x=6, and .*. x=l lb.
3. Let X be the power in lbs.
Then x : 12=3 : 9 ;
. . 9x=36, and .'. x=4 lbs.
4. The capstan is a strong cylinder of wood moveable about a
vertical axis, with short bars, called handspikes, inserted near the top,
by means of which it can be made to revolve. A rope coiled round
the capstan is /ittached to the weight that has to be raised. Hence
mechanical advantage is gained, because the distance of the ends of
the handspikes from the axis is greater than the distance of the coil
of rope from the axis.
Let X be the weight the men can support in cwt
Then6:x=2 :5 ;
.'. 2x~30, or, x=15 cwt.
5. If a vertical line be drawn through (see diagram on p. 98 of
the Statics) and P be made to act at the end of this radius of the
wheel, P will produce no pressure on the axle, aud therefore the
pressure on the axle will in this case be the least possible.
6. Since P and TV are equal, it is plain that
equilibrium can only exist when the moments
of P and W about are equal, and this will be
when the direction of P is a tangent to the ,
axle^ as in the diagram. p ^
Fio. 154.
@
52
KEY TO ELEMENTARY STATICS.
7. Let X be the greatest weight that can be supported.
Then36:a;=l :3;
/. a; =108 lbs.
FiQ. 165.
8. Let TFi and TFj be the two weights.
Let be the real centre of the wheel,
and A the axis of the axle.
Then P x AM= W^ x AD,
andPx^iV=Tr,x^^;
.\P,{AM^-AN)^W^y^AB^W^xAD^
.-. 2P X 0M= (TTi + W^jAD.
But if and A coincide,
2PxOAf=2fr.^D;
9. The greatest weight will be supported when W acts at the
extremity of a diagonal, and the least when w acts along a side of the
square.
p : W^\ diagonal of square : radius of wheel,
F :w^\ side of square : radius of wheel ;
.'. TT: to = diagonal of square : side of square
= V2 : 1.
Examples— X (p. 105.)
1. Let 26 be the angle between the strings.
Then2P,cos^=Tr,
and .'. 2P. cos^=P, and cos^= J, or, ^«60*,
and .•.2^= 120*.
KEY TO ELEMENTARY STATICS. 53
2. If If? be the weight of the pulley,
.*. if to be not less than P,
W is not greater than P, and there can be no advantage.
3. P: Tr=l:2cosd
= 1 : 2cos30*=l : 2. 51^-1 : V3.
4. P : 8=1 : 2», and /. P=l lb.
5. Let X be the weight of each pulley.
Thenll = V + |- + f+|,
or, 88=32 + 7x, and /. a=8 lbs.
6. P-f+f +f +f ;
.-. 8P«6 + 12 + 6 + 3, and /. P=3J lbs.
.-. P=10 + 2, or, P=12 lbs.
8. P=|+i + i+i;
.•.P=y=2, or,P=21bs.
^ Ty W W W
.-. 8P=7TF, or, P : TF=7 :8.
10. Let X be the weight of each pulley in lbs
XXX
Then75-V^ + ^ + ^ + -f.;
or, 63 =56 + 7a;, and .-. a=l lb.
11. Let X be the weight of each pulley.
ThenTr-,^+^ + i^+*-;
8248
or, 8Tr= W+ 7x, and .% ac— TT.
<
54
KEY TO ELEMENTARY STATICS,
12. Let the number of pulleys be n.
Then W=v)+p,
and;7=^^jj^, or, 2*.|>-j?=w;
/. «;=(2**-l)jp, and 2"-l is necessarily an odd number, because
all powers of 2 are even numbers.
13. 2=|^+j+i+J+iV;
or, 32= W+ 8 + 4 + 2+ 1, and .-. W= 17 lbs.
14. Pressure on beam=F + 21^ + 4P'+ . . . 2«-i . P'
=(2'»-l)F;
/. Tr'=(2»-l)P';
also, W=2^.P;
.••5^-5^-2--(2«-l)=l.
15. Let D be the fixed point,
p The beam is acted on by two parallel
forces,
one in BM=P,
one in CN=-2P;
,\ there is equilibrium when
BD:CD=^:P
=2:1.
16. P:8 + 112=1:G;
.-. 6P=120, and .'. P=20 lbs.
17. 2:TF+8=1 :7;
.-. 14= W+ 8, or, W= 6 lbs.
18. Let W be the weight of the man, P the power he exerts.
W
ThenP: -2- = l:7;
W
• p=— .
•• -^ 14
Hence pressure on noor= ^ " TJ == IJ"'
KEY TO ELEMENTARY STATICS.
55
19. P:TF+3P=1:6,
or, (iF= W+3P, and /. TF=3P
20. Let w be the weight of the lower block.
Then3:10 + ir=l :4,
or, 12«lO+t0, and .*. w=2 lbs.
21. Let n be the number of pulleys aD the lower block.
If the string be fastened at the upper block, the number of strings
at the lower block is 2n,.
If the string be fastened at the lower block, the number of strings
at the lower block is 29i + 1.
In the first case, p : w+p=l : 2n, and .'. w= (2n— l)p.
In the second case, jp : w +jp = 1 : 2» + 1, and .". w^^np.
•K
22. Let D be the point of suspension.
Then, referring to the diagram on page 104 of the Statics^ the action
of the strings on the bar will be represented by ,p gp p
three equidistant vertical forces, P, 2P, 4P.
Let 2a be the length of the bar, x the distance
of D fipom the force 4P.
When there is equiUbrium, the bar being ~
horizontal, taking moments round D,
4P X x=2P X (a-jr) + P X (2a-a;),
or, 4a;=2a-2a;+2a-a;; W
.•. 7a; = 4a, or, a; = fa. Pio. iss.
23. Taking the diagram on page 104 of the Statics, extended to
six pulleys, but omitting the string RA, and taking w for the weight
of each pulley,
tension of first string, SB =w,
tension of second string, TC=2w^
tension of third string =7w,
tension of fourth string = 15m7,
tension of fifth string = Zlw ;
/. w+Zw + 'Jw + lbw + Zl'tr^W I
.-.w: W^=l :57.
56 KEY TO ELEMENTARY STATICS.
%
Examples — XL (p. 112.)
1. Since 62 + 122=169=132.
F : IF= height of plane : leogth of plane
=5:13;
.'. P= -rr- tons=l^ tons.
2. Since 32 + 42=25=52,
W : 12= length of plane : base of plane
=6:4;
... U=12;^lbs.=81bs,
3. W will be represented by 6 ;
/. ie=}fF, andP=fTr.
«
4 The height of plane will be represented by 3.
P : Pr= height of plane : base of plano^
P: 12=3:4;
.-. P=af lbs.=91bs.
6. P : P= height of plane : length of plane ;
.'. 1 : 2=height : length ;
.'.1:2: iv^3= height : length : base.
Then P : fr=height : base
.-. TF=V3 lbs. ;
and if ^ be the inclination of the plane,
sin^= J, or, ^=30^
6. Let P and P' be the two forces,
height : base=P : W
= 15:20=3:4;
.*. height : base : length =3 : 4 : 5.
Hence P':20=3 :6;
.-. P'=12 IbB.
KE Y TO ELEMENTAR Y ST A TICS, 5 7
7. The resistance to motion arising from the rails is (50 x 10) lbs.,
and therefore the tension of the rope is thus relieved to the extent of
600lb8.
Also P: 50=1 :30;
.'. P= J8 tons ;
/. strain on rope =(jj -5*55%) tons=l§}f toms.
8. P : }F» height of plane : base of plane,
and .*. if a be the inclination of the plane,
tano=^=r-^=V3> a^d •*• «=60*.
9. P : fTss height of plane : length of plane ;
/. P: 2^2=1 :V2;
.-. P=2 lbs.
10. P : Tf™ height of plane : base of plane ;
.-.P: 56=1:1;
.-. P=56 lbs.
Hence any force greater than 56 lbs. will move the weight.
11. P : TFb height of plane : base of plane ;
.-. P:12=V3:1;
/. P=12V31bs.
12. P : W^ height of plane : base of plane,
and the angle of inclination is 30*^ ;
.-. P:10=l :V3;
. p 10 „ 10^3,,
.*. F—'-pz lbs. =s — ^ lbs.
13. Let P and 2P be the forces.
Then -^ ^heig ht of plane
W length of plane'
^^ 2P^height_of_plMie .
W base of ^lane '
. 1 __ base of plane
length of plane *
,% angle of inclination is 60^.
58
KEY TO ELEMENTARY STATICS.
14. Let P be the tension of the string A W,
A Then P : W^= height : length
W
.-. P=
72*
Hence pressure on the ground
Pro. 159.
=Tr-
V2
-'KS^)-<-^')
15. P : fr=height : length ;
/. P:4=«l:2, or, P-21bs.
W
Fig. 160.
P 16. Let I POD^ L ROD.
Draw DM parallel to OF,
Then the sides of MBO are parallel to P, W, R,
Also z 3fD0= z DOP
= z DOM.
Now P:i2=DM:0M.
But DM^OMy and .-. P=JK.
17. Let Pi be the power acting parallel to the plane, Pa the power
acting horizontallj.
Then Pi : TT^ height : length,
and Pf : TF= height : base ;
.*. Pi X length = Pa X base.
Now, the length is greater than the base ;
.*. Pi is less than Pg.
Ik
18. The string will just support the weight when
height of plane : length of plane =10 : 20
= 1:2,
and in this case the angle of inclination is 30^
.'. as soon as the inclination is greater than 30**, the height of the
plane is increased, a power greater than 10 lbs. is required, and the
string breaks.
KEY TO ELEMENTARY STATICS,
59
19. The tensions of the parts of the string A W and A W are the
same^ represent each by P.
.W AG
*"^p-=Xi>'
. W AB
"W^AC'
j/J^
B DC
Pio. 161.
20. The power acts first along the plane and then horizontally, and
taking 2a and a as the angles of inclination, and 27rand 3Was the
weights supported in each case.
;.= tana ;
.*. 28in2a=3tana;
3sina
.'. 4sina.cosai
cosa
.*. cos*a=-^-, or, cosa=^ ;
.*. 0=30", and .*. 2a=60*.
21. Let a be the inclination of the plane.
10 height
^**^Tr=leigth^
, 20 height
*°*^TF==-bSi"^
._b^e_ ,,
"length ^^ ^'
.*. a is an angle of 60*.
22. Height of plane : length of plane = 12 : 20=3 : 5 ;
.'. height : length : base = 3:5:4.
Naw, horizontal force : 20=3 : 4 ;
/. horizontal force = 15 lbs.,
and 15 : 12=5 :4
i
6o
KEY TO ELEMENTARY STATICS.
Again —
Pressure in first case : 20=4 : 5 ;
.*. pressure in first case =16 lbs.
Pressure in second case : 20=5 : 4 ;
.'. pressure in second case = 25 lbs.,
and 25: 16=52; 42.
23. Height : length : base =1:2: V3,
and hence the angle of inclination is 30'
24. ^C: ^0=3:5;
Hence if AB become the height of the plane,
3 lbs. : weight supported = 4:5;
.'. weight supported = 3| lbs.
25. The force required to keep W at rest, acting along the plane,
is W , sina.
Hence the force acting down the plane must be W . tana - W , sina,
for then the resultant force tending to pull the body upwards along
the plane will be
TTtana- (TF". tana— PT. 8ina) = TF. sino.
26. There will be equilibrium when P and W make equal angles
with the direction of 12, each angle being evidently a, the angle of
inclination of the plane.
7
Y\
sm2a sma
and .*. Bi=2W. cosa.
^c
But when P acts parallel to the plane.
y
f
B=W.coaa.
\
Fia.:
V
162.
Hence Bi=2B.
KEY TO ELEMENTARY STATICS. 6i
Miscellaneous Exercises (p. 121).
1. 0-4, ^the direction of the re-
sultant of the equal forces P and Q,
bisects the angle TO(i,
OA is opposite the resultant of the ^ < ^iy^ > A
equal forces R and iSf, and if OB be
tihe direction of this resultant^ it bi-
sects the angle BOB, R Fio. 16S.
/. l^AOq, QOB, BOB ==2 rt. z-= z AOP, POS, SOB ;
.-. iQOB^iPOS.
The four forces are not necessarily equal, for the magnitude of the
resultants of each pair, depending partly on the magnitude of the
forces, and partly on the angle between the directions of the forces,
may be equal, without equality existing between the four forces.
2. P2 + Qa + 2Pg.cos^=iJ2=p2+g2 + 2Pg.cos(45"-^ ;
/. cos^=cos(45*-^);
.'. cosd=cos 45' . cos^ + sin 45' • sin^ ;
.*. cos^= -T^ . cos^ + -jT . sin^ ;
. ". V2 . cos^ = cos^ + sin^ ;
/. i/2=l+tan^, or, tan^=V2-l.
3. From A in BO produced draw AB parallel to P, meeting 0(i
mB.
P
^a
Then if in magnitude,
22, Q, P are in descending order ;
AO^ OB, BA are in descending order ;
I OB A, L BAOy L AOB are in descending order ;
.*. L POQf L BOB, L BOQ are in aaceindii\\^ ^TdkS^.
62
KEY TO ELEMENTARY STATICS.
4. ML, NO, AC axe equivalent to MP, MB, ND, NP, AG
which are equivalent to CN, ND, CM, MB, AC,
which are equivalent to CD, CB, AC,
which are equivalent to CA, AC,
which represent forces in equilibrium.
D N C
Fia 166.
5. Let h be the neight of the steeper plane, V the base, a the
length, Pi and P% the powers employed.
Now since the power on the steeper
plane is equal to the resistance on the
other, the height and base of the second
" are respectively equal to the base and
height of the first.
Then since three times resistance on
steeper = resistance on the other.
a a '
.-. 9{a^-h?)=-h\ or, Za^sJlO.h.
Then Pi : W^h : a=3 : >/10 ;
p, : W^>Ja^^n? : a=^ :3a=l : VlO.
6. Let be the given point within or without the triangle, O the
centre of gravity
A A
w
C B
Flo. i«r.
KEY TO ELEMENTARY STATICS.
63
Then GO is the resultant of QB, BO,
GO is the resultant of OC, GO,
GO is the resultant of GA, AO ;
.-. three times GO is equivalent to GA, GB, GO, AO, BO, CO.
Now GA, GB are exactly counteracted by GO, for if we produce
CG to D, the middle point of AB,
GA, GB have a resultant = 2^2) =0^ ;
.'. three times GO is equivalent to ^0, BO, CO.
7. Let E and be the centres of gravity of AMN and MBCN,
G tiie centre of gravity of ABC.
Let^O=x, ^i>=^.
Now area of MBCN =3 times area of AMN ;
.-. OGxZ=BGxl,
Now OG=x-^
3
BG^AG-AR=%h-'i.^=h.;
.-. 3»-2^= A and /. x=^—h.
«i
8. Let AC=3 feet, CB=4feet, AB^b feet.
Then z JLC^ is a right angle.
Draw DN parallel to AC.
Let ti, ^ be the tensions of CB, CA.
Then CN :ND :DC=:ti:t^:25.
But, by similar triangles,
CN:ND:DC=^AC:CB:BA
=3:4;5;
/. ei:<a: 25=3:4:5;
.'. ^=15 lbs., and ta»20 lbs.
«5
Fio. 169.
64
KE Y TO ELEMENTAR Y ST A TICS,
9. The direction of W^ the weight of the rod
BG^ bisects the angle BAC,
The tensions of AB^ AC are equal, let each
be represented by t.
Then W^=^t^ + t^ + 2t.t ,coaBAC
=<2 + t2 + 2i^cos60"
=«2 + t« + 2i«.{
.•. tension of each string sbs-t-=—--^ .
R
Pio. 171.
10. Let be the point on which the forces P, Q, R acting are at
rest
Then POiJ = (ISO* - 30") = 150'.
And B:Q= bulPOQ : ^inPOJS ; (Art. xxxix.)
.•.V3:l=sinP0g:J;
.-. 8inP0g= ^ ;
'. P0g=60% or 120".
■*^
Hence ^OU- (360^ - 150" - 60") = 160', . .
orC01J=(360;-160"-120")-=90"; . .
.-. P : Q- 8inl50" : sinlSO", and /. P=ft
or P : §- Mn90" : sinlW, and .-. P=2Q.
Fio. 1.
Fio. 2.
KEY TQ ELEMENTARY STATICS.
65
11. First, when P acts horizontally
Next, when the direction of P makes an angle of 30** with the
plane, resolying in the direction of the plane, there is equilibrium
when
P.cos30**=Jr.cos60°,
P cos60-*_l .V3_ 1
W^ cos30'"" 2 • 2 "y3'
which is the relation between P and W established in the first case.
P
Ri R2 /
B
y]
W
Fig. 172.
Again, if i^ and i^ be the pressures on the plane,
B^^ sin90^ _ sin90° ^g
P sinl50* sin30'
J^ 8inl50^ _, .
P " sinl50" '
12. P : y : P=sinl20' : sinl05" : sinl35'
(Art. xxxix)
P
=sin 60** : sin 75^ : sin 45*
_>/3.V3 + l .JL
2 • 272 V2
^V6 . V3 + 1 2_
2V2 • 2^2 • 2^/2
=V6:V3 + 1 :2.
E
/05*
¥\<a. \1^
\
66
KEY TO ELEMENTARY STATICS,
Fio. 174.
13. Let ^0 be the boat, AB the mooring
chain, G the stake by which the rope JC is
B fastened. Let the current act in the direction
DA.
Now, suppose AB to be cut. Then the boat
is acted on by two forces, one acting along -40,
the other along ACy and the resultant of these
forces acts towards the bank GB^ to which the
boat will therefore swing
Fig. 175.
14. Let oAy oBy oC represent the
forces.
Join AB, BG, GA.
Complete the parallelograms AoBF,
BoEGy AoCD,
E Then CJAoBF^ EJFoBE
(Euclid, L xxxvi.)
=OoBj&(7;
.-. i^AoB= lBoG.
Similarly, each of these triangles
= A AoG,
15. ABy BChave a resultant
AG.
.'. ABy BG, DG are equiva-
lent to AGy DGy of which the
resultant is 2EGy t.e. EGF.
Fig. 17a
KEY TO ELEMENTARY STATICS.
67
16. Since the figure is symmetrical, pressure on B
is equal to pressure on 0.
Let P be the pressure on each of these pegs.
Then the tensions of jBD, GJD will each equal P, g
and zPDO= 60°.
.-. Tr2=p3 + p2 + 2P.P.cos60°«3P2;
.-. P=
-^, or,P^ T-'
Also, pressure on JL is clearly equal to W.
17. The three forces acting on the sphere are
the tension of the string AO, the pressure of the
wall acting along the radius PO, at right angles
to ABf and the weight of the sphere acting
vertically through 0, and the lines of action of
these forces are parallel to the sides of the tri-
angle AOB,
,\R:T:JV=OB:OA:AB
W
Pio. 178.
18. Let T be the tension of the string, a the inclination of each
plane, B the pressure on each
plane, acting as a normal to the
sphere.
Resolving vertically, for the
equilibrium of the sphere,
pr=2P.cosa.
Resolving horizontally, for the
equilibrium of either plane,
T=P.smo;
.•.T=tana.-2. = y.^.
68
KEY TO ELEMENTARY STATICS.
1 9. Let ^ (7 be the post, B its middle
point, N^
I DBC= 30', and . •. z BBC= 60^ ^^
The angle between the two sets of
telegraph wires is 120**, and if iS be
the resultant of the equal tensions of
the wires, each = T,
222^ ^12 + >zi2 + 22^ . coslSO" ;
. 222=3T2 and/. i2=T.
Let t be the tension of rope BI).
Then, taking moments round (7,
i2x^(7-«xJ5a.cos60°;
AG
.-. TxAC=t X -^ X J, and .-. t=4T.
Fra. 180.
\
20. Let ABCD be the quadrilateral, and BF=CE, DG^AE,
Bisect BC in 0. Join OA, OG, OB,
Let H and K be the centres of gravity of
ABG and BBC, Join B^ cutting OG in i.
Then '.• OH^lOA, and OK^iOB ;
.'. fix is parallel to AB ;
.', OL=iOG, and L is the centre of gravity
of triangle EFG, Draw AP and DQ per-
pendicular to CB,
Then triangle^JBO: triangle DC© = ^P : DQ
=AE:BE
=BG:AG
=KL:LH.
Therefore the triangles ABC and BOB will C
balance about Z, or the centre of gravity of
the quadrilateral is L ; and this is also the centre of gravity of the
triangle EFG,
21. Let A, B, C be the three pegs.
z 5^0=120*;
.-. z ABC= L AGB^W.
Then pressure onA^2W. cos60'=Tr,
pressure on B^2W, cos30'= fT. ^3
= pressure on C,
Fio. 181.
KEY TO ELEMENTARY STATICS.
69
The yertical pressure on ^= TF acting npwards.
The yertical pressure on both £ and C
= PF+ W , co860*=-— acting downwards.
The reactions which support the weights will
be equal and opposite, so that we shall haye, in
order to support the wei^ts, —^ at both B and C'
acting upwards, and W acting downwards at A ;
or 3 TT upwards, and W downwards, or 2 fF acting upwards, which
will just support the weights.
22. Resolving along Oa; and Oy, of which Or coincides with BOE^
observing that co660'= J and sin60'*=-^
Z=6 + 6.J-l.i-2-3.J + 2
=6+3-J-2-|+2=6
F=6 ^ + 1 ^
3.^g-4.4^=0.
2 2
Henoe the resultant acts along OEy and its magnitude is 6 lbs.
70
KEY TO ELEMENTARY STATICS.
23. Let M be the middle point of the rod, W the weight of the
^ _ rod, T the tension of each of the strings
f AD, BE.
J Take moments about C, and
'^ T,AC=T.BC+W,MC;
_ or, T. {AM+ MC) = T. {BM- MC) + W,MC;
B and /. since T . AM=T,BM
T.MC=W.MC-T.MC;
M
V
w
Fig. 184.
W
24. Let ABC be any one of the triangles, and
AC, CB represent forces tending from A, and AB
a force tending towards B, it is plain that the re-
sultant of AC, CB will be represented by AB, and
.'. the resultant of AC, CB, AB will be represented
Fio. 185. by 2AB,
25. Since the angles AEB, ABB are right angles, a circle described
on AB as diameter passes through E and D ;
.-. I ABE= L ad;e,
and since the angles AFC, ADC are right
angles, a circle described on ^ C as diameter
passes through F and D ;
.-. z ACF=^ L ADF.
Now z ABE^ L ACF,
by similar As ABE, ACF;
^ .-. I ADE= I ADF,fixA.\ J jD bisects z FDE,
*. forces are equal
Fio. 186.
»
Fio. 187.
3 26. Let AB, J.C be the arms of the lever.
Let the forces P, P act along PD.
Then the pressure on A will be represented by AD,
Now ^jD2=2Pi>2 ;
.-. ^2)=V2.Pi>,
or pressure on A^ ^2 » T*.
ELEMENTARY HYDROSTATICS.
KEY.
Examples — I. (p. 8.)
(1.) 1 J : 64 = 1 ton : weight supported ;
.•. weight = — - — tons = 56f tons.
(2.) IJ : 240=3 cwt. : weight supported ;
/. weight =^^^:i^^-:i^ cwt. =600 cwt. = 30 tons.
(3.) Area of small piston : area of large piston = (1})2 : (60)*
=9x9:50x60x64
.-. weight supported ^ 50x50x64x 15 ^^^^^ ^ 800000 ^^^
= 29629-S2&lbs.
(4.) Let X represent the pressure in lbs.
Then2:144=a;:4j;
2x9 1
/. x=---- — x=r7: , .'. pressure will be 1 ounce.
144 X 2 16
(6.) Let % represent the pressure in lbs.
Thenl :144=a;:9 ;
9 1
.'. «=——=—, .•. pressure will be 1 ounce.
144 16
I
72 KEY TO ELEMENTAR Y HYDROSTATICS.
(6.) Area of horizontal section of large cylinder = = '
sqnare inches.
Let X s measure of lifting power in cwts.
mu 11 10x10x22 on
Then If : ^=20 :x;
•*• **^ — 5q — cwt. =s 6687 J J cwt.
Examples— II. (p. i8.)
(1.) Pressure at 42} feet ^%^ • pressure at 32 feet
^^® X 15 lbs. =20 lbs.
32x3
(2.) Pressure at 20} feet ==^ • pressure at 8 feet
^^ x^ lbs. =37 A lbs.
^
8x2 3
(3.) The pressure at a depth of 8 inches in the second fluid Is
double the pressure at a depth of 4 inches, and is therefore equal to
the pressure at a depth of 6 inches in the first fluid.
.'. the pressures are as 7 : 6.
(4) The pressure at a depth of 12 inches in* the second fluid is four
times the pressure at a depth of 3 inches, and is therefore equal to
the pressure at a depth of 8 inches in the first fluid.
.'. the pressures are as 9 : 8.
(5.) Let X be the height of the column in feet. Then (30 - x) feet
is the depth of the top of the column, and 30 feet is the depth of the
bottom of the column.
.-. 30-x:30=2:3,
or, 90 - 3x=60, and .'. «= 10 feet
6.) Pressure at 12 feet — \\ pressure at 15 feet
= if X 15 lbs. = 12 lbs.
(7.) Pressure at 12 feet = ^ pressure at 4 feet
= yx31bs.=91b«.
KE Y TO ELEMENTAR Y HYDROSTA TICS. 73
(8.) Cubic content of water =(20 x 2J) cubic feet = 50 cubic feet ;
.'. pressure =(50 x 1000) ounces =3126 lbs. = 1 ton 7 cwt. 3 qrs. 17 lbs.
(9.) Cubic content of mercury = (8 x 3) cubic inches ;
13600x24 1700 n iv los
.*. pressure = — :r=^ — ounces =—x— ounces=ll lbs. 12§ ounces.
(10.) Cubic content of water = (24 x 18) cubic feet ;
.-. pressure =(24 x 15 x 1000) ounces =22500 lbs.
(11.) Let ABC represent the base of the cistern : AD a perpen-
dicular on BC, C D B
Thsm AD^BB . ^Z.
Hence, if each side of the triangle be 6 feet,
area of base=^D . BD^QaJS ;
.'. cubic content of water = (9 ^3 x 2) cubic feet ;
, 9V3x2xl000„ -,__ ,-„
.*. pressure on base=-^^^^ r^ lbs. = 1125^3 lbs.
(12.) When the spout has been broken midway, its upper extre-
mity is three-fourths of the height of the tea-pot, and therefore this
is the height to which the tea-pot can be filled.
(13.) Since the external pressure on the cork increases as the bottle
sinks, while the internal pressure on the cork is constant, the cork
will be forced in when the external pressure exceeds the internal
pressure.
(14.) Pressure on bottom
= (112 X 4 X 1000) ounces= Y^^Y^^3 tons = 12 J tons.
(15.) Let X be the height of the water in the tube in inches.
The pressure produced by this on the base of the piston, which is a
square inch in area, will be equal to the weight of x cubic inches of
water, which has to lift 7 lbs. 13 ounces, or 125 ounces ;
X X 1000
1728
= 125
1728x125 «,^. , ,«- ,
3?= — j^TQTr — = 216 mches=18 feet.
74 KE Y TO ELEMENTAR Y HYDROSTATICS.
Examples — III. (p. 27.)
(1.) 1 cubic foot of copper weighs 8*91 x 1000 ounces, or, 8910 ounces.
1 cubic inch of copper weighs r^^^ ounces.
512 cubic inches of copper weigh — Tna^ — ounces, or, 165 lbs.
(2.) 1 cubic inch of iron weighs as much as 38 cubic inches of
amber.
.*. specific gravity of iron : specific gravity of amber = 18 : 1.
(3.) 1 cubic foot of mercury weighs 13500 ounces.
1350C
1728
1 cubic inch of mercury weighs ounces, or, 7} J ounces.
>
(4.) 1 cubic foot of the substance weighs 50 lbs.
1 cubic foot of the substance weighs 800 ounces,
and 1 cubic foot of water weighs 1000 ounces ;
.'. specific gravity of the substance is '8.
(5.) 1 cubic foot of cork weighs '24 x 1000 ounces =240 ounces.
1 cubic inch of cork weighs r^^ ounces.
200 X 36
36 cubic inches of cork weigh —ry-— ounces, or, 5 ounces.
(6.) A cubic foot of the substance weighs 3000 ounces.
A cubic inch of the substance weighs yfoQ ounces, or, If .| ounces.
(7.) 1 cubic foot of the body weighs — lbs.
m
1 cubic foot of the body weighs - ounces,
and 1 cubic foot of water weighs 1000 ounces ;
/. specific gravity of body : 1 = — ; 1000 ;
m
.-. specific gravity of body = -^^^ = '^^.
^ ^ ^ ^ 1000m m
KEY TO ELEMENTAR Y HYDRO STA TICS. 75
(8.^ 1 cubic inch of iron weighs 4} ounces,
1 cubic foot of iron weighs (4J x 1728) ounces, or, 7776 ounces ;
.*. specific gravity of iron : 1 = 7776 : 1000 ;
.*. specific gravity of iron = 7*776.
ft'7ft V 1 ft
(9.) 1 cubic foot of the wood weighs — — — ounces, or, 1166*6
1^
ounces;
.*. specific gravity of the wood : 1 = 1166 '6 : 1000 ;
.'. specific gravity of wood = l"l5.
274-*? X 1 fi
(10.) 1 cubic foot of ash weighs — -- — -— ounces, or, 844 ounces.
.'. specific gravity of ash : 1 =844 : 1000 ;
.'. specific gravity of ash = '844.
(11.) Let V be the volume of the metal, and b the specific gravity
of the compound.
Then -|- is the volume of the alloy ;
.-. vxl5 + |-xl2=rv +-|-)^«
.-. 15 + 6=^, and.*. «= 14.
(12.) Let 8 be the specific gravity of the lump.
As
Then --- is the specific gravity of one metal (for 2 : lj=4 : 3),
and -^ is the specific gravity of the other (for 2 : 2J^=4 : 5).
Let Vi and v^ be the volumes.
Then Vi x -^ + vg x — = (vi + r^) x « ;
3 5
.-. 20Vi + 12Va=15t;i + 16«a;
.'. 6vi=3va, or, Vi : t7a=3 : 5.
Hence to form 2 cubic inches of the compound we must put | cubic
inches of first and { cubic inches of second.
76 KEY TO ELEMEJSITAR Y HYDRO STA TICS.
(13.) Let Vi and v^ be the yolumes, tOi and w^ the weights.
Then VbVi + Zv^=\vi-\-Vt} 2-6,
or, 15vi + 30t;j= 26t?i + 25^2 ;
and /. Vi : «;j=6 : 10=1 : 2.
^^^'''r6"*"¥""2^'
lOtOi Wa IOWi+IOk^
'''"15"'*"3"= 25 '
or, 60wi + 25u?a = 30Wi + GOwj ;
.*. 20wi=6tr2 ;
.*. tPi : Wa=5 :20=1 :4.
(14.) Let a; and y be the measures of sea-water and fresh water
respectively.
Then x x 1*027 + y x 1 == (sc + y) x 1*009,
or, l*027a; + y=l*009a; + l*009y,
or, -OlSx^'OOOy;
.'. X :y=l :2;
that is, proportion of fresh water to be added = 2:1.
(15.) Let w be the measure of the weight of each substance, d the
density of the compound.
mi,^„ w , to 2m;
„ 100^100_2
'325'^276~d'
(16.) Let V be the measure of the volume of each substance, s the
specific gravity of the compound.
Then v X 2*5 +t7 X l*5 = 2v X « ;
.*. 2*5 + 1*5=2«, and .•. «=2.
(17.) Let s be the specific gravity of the compound.
Then 5 x 11*35 + 5 x 7*3= 10« ;
or. 11*35 + 7-3=2«, and .*. 8=9*325.
\
KEY TO ELEMENTARY HYDROSTATICS. 77
(18.) Let V be the measnie of the volume of each fluid, (2i, 4i9 (2s the
measures of their densities, d the measure of the density of the
mixture.
Then vii + t;(2s + 1;(^ ss (i; + 1; + 1;)({ ;
.'. (ii + d8+(t=3(i, and.'. (i,=3<i-(ii-i».
(19.) Let « be the specific gravity of the compound.
Then 10 x 8-9 + 7 x 7-3=17«,
or, 89 + 51-l = 17«;
.-. 17«=140-1, and .'. «=8-241 . . .
(20.) Let % be the specific gravity of the mixture.
Then^ + A + l=^-±6±7
•7 -8 -9 B '
^^ 60^15^70 18
.•.?^«l?,and.-.«=:-802 . . .
126 «
(21.) Let % be the specific gravity of standard gold.
Then 11 x 19*3 + 1 x 862= 12« ;
.•.212-3 + 8-62=12* ;
.-.«= 18-41.
(22.) Let « be the specific gravity of the mixture.
Then 63 x 1-82 + 24 x 1 =86«,
or, 114*66 + 24=86* ;
.-. «=1-61 ...
(23.) Let 9 be the specific gravity of the compound.
Then4x2 + 6x3 + 6x4=(4 + 5 + 6)«,
or, 8+15 + 24=15*;
/. *b3-1S.
(24.) Let 10 be the weight of gold in ounces.
Then _!g__^ ll'5-u7 ^ir5
19-35^ 2-62 7-43'
^, 100M7 1150-100117 1150
' 1935 ^ 262 743 '
20W 575-50117 ^1150
' 387 ■*■ 131 743 '
• • —
262 0?g + 222525 - 193 50M7 1150
50697 " 743'
78 KE Y TO ELEMENTAR Y HYDROSTA TICS.
or.
222526 - 16730tr 1150
' 50697 "" 743 '
or, 165336075 -12430390tr=58301550;
.-. 107034525 =12430390u7,
and /. t(;=8'6 . . . ounces.
(25.) Let tTi, w^ be the weights of iron and gold, Vi, v% the volumes
of iron and gold.
Then 7-8t?i + 19-4*,«8 (vi + t;,),
or, ll'4t?a=*2t7i;
.*. t;i:t;j=ll'4:-2
=57 : 1.
Again, 15. + J£!-=^_±i^,
^^ '7-8 19-4 8 '
'''^' 39"*'W"~"8~'
, 485ti7i + 195wa _ tPt + tOg
•'• 3783 8 '
or, 3880k?i + 1560i«a= 3783wi + 3783t«?» ;
.-. 97«\=2223wa;
•^ tri:w2=2223 :97.
ExAMPLBS— IV. (p. 42).
(1.) Weight of water displaced =^ of weight of glass ;
.'. specific gravity of water=T^ of specific gravity of glass ;
.*. specific gravity of glass = V* = 3'S.
(2.) Pressure
= weight of {(28 x 1760 x 3) x J x 480} cubic feet of sea-water.
28 X 1760 X 3 X 480 X 1-026 x 1000
ounces
4
7 X 1760 X 3 X 480 X 1026
tons.
16x112x20
(110 X 9 X 613) tons=507870 tons.
KEY TO ELEMENTAR V HyDROSTA TICS. 79
(3.) Let a;=the part immersed, t7= volume of body,
Thena;:v=3-3:4-4;
/. x—-r-. v= 77 v=a -J- of whole body.
4*4 44 4 ^
(4.) Let x be the number of grains a sovereign weighs in water.
Then 122'5 - a: = weight of water displaced ;
.M9-4 : 1 = 122-5 : 122-5 - ar,
or, 2376-5 - 19-4a;= 122*5 ;
2376-5-122-5 .,-,« . . ._^ o^io
• • *■* — "To^i =116Jf grams =4 dwt 20Jf grams.
(5.) Weight of water displaced =2 lbs. ;
.'. specific gravity of substance : 1 =8 : 2 ;
.'. specific gravity of substance =4.
(6.) Let X be the weight required, w the weight of the tub.
Then since weight of water displaced by the whole tub=s4«;,
a;=4w-tt;==3w— three times weight of the tub.
(7.) Part immersed : whole body =1*4 : 2*1 ;
14
gT of whole body— ^
14 2
.-.-part immersed=— of whole body =-5- of body.
(8.) Weight of water displaced by leads=YT:^ ounces ;
. •. pressure on bottom = ( 1 - yrri ) ounces = - - ounces = r= ounces.
(9.) Half a cubic inch of water weighs y=^ ounces,
a cubic inch of cork weighs 3-;=^ ounces ;
17^0
• 1.x X V jj J 500-240 65
.*. weight to be addea= — - ounces=— ounces.
(10.) Since the specific gravity of the cork is one-fourth of the
specific gravity of water, weight of water displaced by the cork = four
times weight of the cork.
.'. tension of strings three times weight of the corks 3 ounces.
8o KE Y TO ELEMENTAR Y HYDRO STA TICS.
46
(11.) Weight of water displaced by lead—rrig ounces ;
.-. tension of string= ^46 - —^ ounces* ^46 x ^l"^ ~ ^ \
ounces
0/ \ ll'O /
46 X 105
115
ounces » 42 ounces.
(1000\
•24 X Ytoq) o^^ces=^^^ ounces ;
240
1728
^ , , . , , . 240 5
.'. we must add a weight of p^Ho ounces, or, 5^ ounces.
(13.) Weight of water displaced ==| of half the weight of the cube
ss j of weight of the cube ;
.% tension of string ={ weight of cube={ lbs.
(14.) Weight of water displaced = (42 - 30) ounces = 12 ounces ;
.'. specific gravity of substance : 1&=42 : 12 ;
42
.*. specific gravity of substance «=—= 3*5.
(15.) 14 lbs. « (14 X 16) ounces=224 ounces.
.'. weight of water displaced =(2560 -224) ounces =2336 ounces ;
.'. specific gravity of substance : ls=2560 : 2336 ;
.'. specific gravity of substance=-— 5-=— -.
2336 73
(16.) Weight of water displaced by substance
= (20 + 12 - 18) ounces= 14 ounces ;
.'. specific gravity of substance : 1 = 12 : 14 ;
12 6
.'. specific gravity of substanceos--— __.
14 7
(17.) Weight of water displaced by mahogany
« (375 + 380 - 300) grains=455 grains ;
.*. specific gravity of mahogany : 1 = 375 : 455 ;
.'. specific gravity of mahoganyso -^a:-- .
KE V TO ELEMENTAR V HYDRO STA TICS. 8 1
(18.) Weight of water displaced by metal
=(120-113) grains = 7 grains;
.'. specific gravity of metal : 1 = 120 : 7 ;
120
.'. specific gravity of metal = ,,- ■" 17 J.
(19.) Weight of water displaced by spar
= (190 - 120) grain8=70 grains ;
.•. specific gravity of spar : 1 = 190 : 70 ;
/. specific gravity of spar = . = — =r2f •
(20.) Weight of water displaced by former body
= (4 + 3 — 2^) ounces = 4| ounces ;
/. specific gravity of former body : 1 =4 : 4| ;
A. 1ft
/. specific gravity of former body=-^=-- .
(21.) Weight of water displaced by wood and lead
= (22+12-8) lbs. = 26 lbs.
Weight of water displaced by lead
22 „ 2200 „ 440 „
"n-35 ^^«- = 1135 ^^'- =227 ^^'•'
.•, weight of water displaced by wood= (26 - o3y ) l^s* = 0017 ^^^ 5
5462
.*. specific gravity of wood : 1 = 12 : gg^ J
.. ', f A 12 ^ 227 1362
/. specific gravity of wood= ^^^ ^2731 *
(22.) Let tc= weight of substance in vacuo, 2ia= weight of sinker
in vacuo, «= specific gravity of substance ;
.'. {w + 2w) — 2wy or, w= weight lost by both in water.
Let t;= volume of sinker, then v x 1 = weight of water displaced by
sinker;
/. IP- v= weight lost by substance ;
. w-v 1
• • ^ — J
W 8
VS — V 1 , 8—11 tx
or, — , and .'. ^— ^=— , or «-2.
V8 8' 8 8
i
82 KE Y TO ELEMENTAR Y HYDRO ST A TICS,
(23.) A cubic foot of cork weighs ('24 x 1000) ounces, or 240
ounces ; .'.to displace a cubic foot of water, weighing 1000 ounces,
we must increase the weight of the cork by 760 ounces, or 47} lbs.
(24.) Let x^Yolume of part immersed.
Thenaj:a: + 8=1 :3;
Hence whole length of the cylinder = 12 feet,
(25.) Four-fifths of the cylinder are immersed.
Let 8 be the specific gravity of the cylinder.
Then« : '825=— : 1 ;
5
.•..=il^=4x-165 = m
5
(26.) Because the specific gravity of salt water is greater than that
of fresh water.
(27.) Let X parts of an inch be in the mercury, then \-x parts of
an inch are in the water.
Then ac X 13-6 + (1 - a) x 1 = 1 x 7-8,
orl3-6a;+l-a;=7'8;
/. 12-6a;=6-8;
• fl. 5§.-_3^ inches
..a-— --mcnes.
(28.) Since the specific gravity of the body is half the specific
gravity of water, it will float with 500 ounces of its weight above the
surface.
And since 1 cubic foot of water weighs 1000 ounces,
1 cubic foot of the body weighs 500 ounces ;
.'. 1728 cubic inches of the body are above the surface.
(29.) 12cubic inches of water weigh — ^ - — ounces, or, — ounces,
1728 18
or, 6J5 ounces ;
.'. weight of body in water = 8 lbs. -6J| ounces, or, 7 lbs, ^^
ounces.
KEY TO ELEMENTAR V HYDRO ST A TICS. ^3
(30.) Let X =■ edge of the cube in inches ;
/. 1 X X* = cubic inches of water displaced by the cube sinking
an inch;
1000
• • ^ ^ V70Q~^®^^^ ^^ ^^ water in ounces ;
1728
/. x^x
1000^1000
1728 3
.*. x2=576, and «=24 inches =2 feet.
(31.) Content of box externally = (10 x 8 x 6) cubic inches = 480
cubic inches.
Content of oox internally = (9^x7^x5^) cubic inches rs'ii^
8
cubic inches.
.'. content of substance =f
480 Q— ) cubic inches = -^ cubic
inches.
Hence, if « be the specific gravity of substance,
705« .Q^ , 480x8 3840
— =480x1, or,. =-^^=-^;
/. «=5ff.
(32.) Let DE be the surface of the fluid.
Th&nADEiAFG^l:^,
(Hydrostatics, Art. 63.) ^
But ADE:AFG^A&:AC^. (Euclid, VL xix.)
.-.^52:^02=1:3;
(33.) BE being the surface of the fluid, and J>, ^the middle points
of AB, AC, triangle ADE = one-fourth of triangle ABC.
A B C
Fig. ISO.
B C
Now, part immersed : whole = specific gravity of body : specific
gravity of fluid.
84 KE Y TO ELEMENTAR V HYDROSTATICS.
/. when the vertex is downwards, specific gravity of lamina = one-
fourth specific gravity of fluid ; and when the vertex is upwards,
specific gravity of lamina = three-fourths specific gravity of fluid.
(34.) Let X be the number of pounds in the weight, « the specific
gravity of the fluid, v and -^ the volumes of fluid displaced.
Then 8 + ic=v.«,
and 8=-5-.«;
...®±5=3, ora;=161bs.
(35.) Let « be the specific gravity of the cylinder.
Then -I : 1=8:1;
.-. 5=1 = -75.
(36.) Volume of cube = (^A x A x -I'i cubic feet = ^ cubic feet;
27
,\ _J. cubic feet of the material weigh 2250 ounces ;
o
.*. 1 cubic foot of the material weighs — - ounces •
o
2
.*. specific gravity of the material is — •
Then if a; be the distance in inches to which the cylinder sinks,
a;:3 = 4:l»
3
and .'. x=2 inches.
(37.) Weight of fluid displaced = 1 lb.
Then, if z be the specific gravity of the fluid,
1 :3=«:2'7;
.% 3«=2*7, or«=-9.
(38.) Let v be the volume of the part immersed in cubic inches.
Then v; 1=2-6: 13-5;
.-..= 21^26-
13-5 135
26
.*. the aluminium will sink to a depth of -—- incL
135
KE Y TO ELEMENTAR V HYDRO ST A TICS, 85
(39.) Let w be the weight of the body in pounds.
Then«;:t/; + 2 = — :1 ;
.-. i£?-±_7J», or, 117=4 lbs.
(40.) Let p be the pressure in pounds.
Then 3:3+2?=^ :1,
or3=^?, or,|)=31b8.
(41.) 2 cubic feet of water weigh 2000 ounces.
2 cubic feet of cork weigh 480 ounces.
.*. tension of string =(2000 - 480) ounces =1520 ounces =95 lbs.
(42.) Let «i and «, be the specific gravity of the bodies, s the specific
gravity of the fluid.
Then _!^-=.?L^
tPi — ti; s
(43.) If the specific gravity of the wood be less than the specific
gravity of the water, the water filling up the cavity will cause an
increase in the apparent weight of the bullet. ,
(44.) Weight of first fluid displaced =3 lbs., weight of second fluid
displaced =2 lbs.
If then 81 and s^ be the specific gravity of the two fluids, and s the
specific gravity of the body,
-^=— ,and^ = -;
.'. dividing, -i== —
(45.) Weight of fluid displaced by the body = 1 lb. ;
.'. if « be the specific gravity of the fluid,
« 1 A . 7-7 , ,
777=y>and--»=y=l*l.
i
86 KE Y TO ELEMENTAR Y HYDRO STA TICS.
(46.) Let 8i be the specific gravity of the first sphere, and 8^ be the
specific gravity of the second sphere.
Then 4«i is the specific gravity of the fluid.
Let 2i; be the volume of the first sphere.
Then Zv is the volume of the second sphere,
and 2i; . «i + 3i; . «8=(2t; + 3i;) . 4«i ;
.'. 2«i + 3s3=20«i;
^47.) Let w be the weight in grains of the copper in air.
Then ^--§?7^ 1 .
w 8-85 '
.-. 885w - 887 X 885 = lOOw; ;
.•.785^=887x885;
.-.^=1-56999.
157
Then if 8 be the specific gravity of the alcohol,
. : 1 = ^^^ _ 910 : 15^ _ 887 (Hydrostatics, Art 68.)
157 157
... ,=y69_99.ziM870^14129^.g ^^^
156999-139259 17740 ^
(43.) Let X be the depth in inches to which it sinks in alcohoL
Then a; X -79=4x1 ;
•*• ^''•79''y9 " ^^ ^^^^
(49.) Five lbs. of the compound are immersed. Let x be the
number of lbs. of silver in the part immersed, b-z the number of
lbs. of aluminium in the part immersed.
Then since 10 lbs. of mercury are displaced
X b-x 10
+
10-4 2-6 13-5'
X b-x _ 10
^^'104 ~26"""135'
. 20-3a; ^2 ,
* * 104 27 '
.'. 640- 81x=208, or, a;=4^ lbs. ;
.*. there are 82? lbs. of silver in the whole mass.
KE Y TO ELEMENTAR V HYDRO STA TICS. 87
(50.) Let t«7= weight in pounds of water displaced by the rod.
Then a; : 5 = 1 : 7-8 ;
_50_25
••^ 78""39*
Hence tension of string = no - 55 ) lbs. = 9Jf lbs.
(51.) Weight of water displaced =(1 - "905) ounces = '095 ounces ;
.*. if « be the specific gravity of the silver,
J. = _l =1222.
1 -095 95 '
.-. «=iofy.
(52.) Weight of water displaced by first body = J lb., and, if « be
the specific gravity of the second body, weight of water displaced by
2
second body = — ^Ibs. ;
.*. tension of first string = J lb.,
and tension of second strings 1 2 ) lbs.
Now these tensions must be equal ;
.-. -5-=2 ,and.\ «=»-^=l-8.
(53.) Let X inches be above the surface.
Then9-a;:9=i- :1;
.*. 9 -a;=3, and .'. a;=6 inches.
(54.) If 8 and ^ be the specific gravity of the fluids, V and F' the
parts immersed.
5 : iSr= F' : F. (Hydrostatics, Art 66.)
=m-y :m-x.
(55.) The weight of fluid displaced by body and lead =7 lbs.
Now 5 J lbs. of lead displace J lb. of water ;
.'. 5} lbs. of lead displace 2 lbs. of the fluid ;
.*. weight of fluid displaced by body =5 lbs.
Then if « be the specific gravity of the body,
«:4=3:5;*
12_
D
I
88 KEY TO ELEMENTARY HYDROSTATICS.
(56.) Let i^= weight of the substance in air in ounces.
Th ^~^^ _ sp ecific gravi t y of wate r _ 1
w- 15 "specific gravity of alcohol" *7947'
(Hydrostatics, Art 68.)
.-. •7947w-7*947=i^- 15 ;
.-. 7-053= -205311?, or, v)^^^^ ounces.
Hence, weight of water displaced by substance
/70530 -A
^^Uosa-^V^^"^
50000
= 2053^^^^^'
•*. if X be the number of cubic inches in the substanoCi
lOOOg^SOOOO
1728 2053 '
50 X 1728 86400 , . . ,
••• ^=^053- = 2063 "^^^" "^"^^-
(57.) Let X cubic yards be the volume of the granite, (1 — x) cubic
yards is the volume of the ice.
Then x x 2*65 + (1 - x) x '918=11 x 1 ;
or, 2-65x + -918 - •918x= '92 ;
.*. l-732x=*002;
•*• *= j^3 cubic yard.
866
(58.) 320 ounces =20 lbs. ;
.*. specific gravity of the wood : specific gravity of water =20 : 12.
Hence if x be the part of the wood below the surface,
x:l = 12 :20;
••'^-20- 5
(59.) 3 inches of the wood weigh as much as 2| inches of water ;
.'. 1 inch of the wood weighs as much as | inch of water ;
,•, 1728 inches of the wood weigh -^r — ounces, or, 750 ounces.
KE V TO ELEMENTAR Y HYDROSTA TICS. 89
(60.) Let X parts of a cubic foot be immersed.
Then x: 1=85 : 103;
. „^ '85 ^ 85
1-03 103'
(61.) Let X be the number of feet in an edge of the cube.
Then 7? (x- 100) : x3=-9214 : 1*0263 ;
/. 10263a;- 102-63= •9214X;
.•.•1049x= 102-63;
^. ^^1026300^^^3.3 ^^^
1049
/. x3=936302451-687 nearly.
(62.) Let X represent the pressure.
Then t/; + x : ie7=4 : 3 ;
.-. 3M; + 3x=4ti7, and .*. x=-—»
(63.) The Yolume of fluid, v, displaced is the same in each case.
Hence, if s and tf be the specific gravity of the fluids respectively
V8 :v8'=rri in ;
and .*. 8 \8'—min,
(64) Let V be the volume in inches of the part immersed in water.
Theni; :i; + 4 = l : 3;
.*. 3i?=i? + 4, ori;=2.
Hence, if x be the weight of the hydrometer in grains^
a: + 40 :x=2 :2-:^;
or (a: + 40)x23=24x;
.*. x= 920 grains.
(65.) 980 divisions are below the surface in distilled water^ 954
divisions are below the surface in sea water ;
.*. if « be the specific gravity of sea water^
«: 1=980: 954;
. 980_490_ 13 _,.^272
••'"^954 477^^-^^^^^
90 KEY TO ELEMENTAR Y HYDROSTATICS.
(66.) Weightof waterdisplacedby wood«(13 + 6-8) lbs. = ll lbs.;
/. if « be the specific gravity of the wood,
«:1=:6:11;
.•.«=!= -54.
(67.) Let X be the weight of the hydrometer in grains,
m, « + 67 1
^^^-ir==-866'
.-. •866a; + 58-022«a;;
.-. -1343;= 58022;
68022 ... .
.*. a;= "104- =433 grains.
(68.) B weighs in water (11 - 10) ounces, or, 1 ounce ;
B weighs in air 15 ounces ;
.'. if t? be the Yolume and « the specific gravity of B^
t>(g~ -00 13)^15,
t;(«-l) 1 '
.-. «-'0013 = 15«-15.
.-. 14«= 14*9987;
.-. «= 1*0713 nearly.
(69.) Let t«7 a weight of the substance in air.
The ^~^Q — specific gravity of water _ 1 ^
t/; — 25 ""specific gravity of alcohol"" '7947 '
(Hydrostatics, Art 68.)
.*. •7947w-15*894=w-25;
.*. 9106= -205311;, and .*. '^^-i^^H - 2o) ounces.
Hence weight of water displaced by substance
/91060 ^^\
== I — 20 1 ounces
V 2053 / ""^"^
50000
= -TT ounces ;
2053 '
•*. if X be the number of cubic inches in the substanoe
1000a ; ^50000
1728 2053 '
/. 35=-oQg«- cubic inches =422\f5^ cubic inches.
KE Y TO ELEMENTAR V HYDROSTA TICS. 91
Examples — V. (p. 61.)
(1.) Let the measure of the capacity of the barrel be oe.
Then the measure of the capacity of the receiver is lOx ;
(lOx \®
— I . original density
»0. original density.
(2.) As in Example (1.)
density after two strokes = ( ) . original density
VoX t" X/
= — times original density.
(3.) Let X and y be the capacities of the receiver and the barrel
• ^, / 35 V 729
. _x 9_
•'•x + y~10'
.*. 10a:=9x + 9y, and .'. x ; y=9 : L
(4) No : because the pressure varies with the depth alone ; so that
if the section varied there would still be equal vertical increments of
space for equal increments of pressure.
(5.) Pressure on a square inch = weight of a column of water whose
height is 90 feet and base a square inch + weight of column of mercury
whose height is 30 inches and base a square inch.
= weight of (90 X Yj^ cubic feet of water + weight of 30 cubic
inches of mercury
90 X 1000 , 30 X 47 ,._^ , „... ^^^..
= — :rjT — ounces + — g — ounces = (625 + 235) ounces = 53| lbs.
92 KE Y TO ELEMENTAR V HYDROSTA TICS,
I
P--
(6.) When the mercury stands at Pin the tube,
let be the point at which it stands in the basin.
Now let the mercury fall to 22 in the tube, and
rise to i> in the basin.
Then CD-=^PE;
, Q .*. real fall = apparent fall + ^ of apparent fall
= IJ inches + — of 1 J inches
2 10 **
Fio. 191.
= ( 2" "^ 2o) ^^^^ '^ ^^ inches.
(7.) The mercury would fall to the level of the mercury in the
basin if the hole were made above the column of mercury in the tube.
If it were made in the part occupied by the mercury, it will drive
part of the column up the tube, and the other part down.
(8.) Pressure on of a square inch = 1 ounce ;
.'. pressure on a square inch = 234 ounces = 14*625 lbs.
(9.) Let X be the height of the column in inches.
Then x x 3-4=30 x 13*6 ;
30 X 136
34
x>
oyj_2 ^OKJ ^ j2^ ^^^^ ^ ^^ ^^^
(10.) It will have no effect, because a volume of mercury equal to
that displaced by the iron will descend and allow the iron to take its
place without disturbing the general upper surface.
(11.) The body floats in two fluids, air and water, and the absolute
weight of the floating body is equal to the weight of the displaced
fluids ; hence if the air be removed, more water must be displaced to
support the body, and therefore the body sinks.
(12.) If the air be admitted above the column of mercury it will
drive it down a little, the extent of the fall depending on the quantity
of air admitted.
KEY TO ELEMENTAR Y HYDROSTATICS* 93
(13.) A column of water 20 feet high represents a pressure of
— of 15 lbs. on the square inch.
34
.*. if z be the area of the valve in square inches,
20 X 15 X a; ,^ , ^, „
«7 lbs. = pressure on valve = 21 lbs. ;
34x21 714 ^_
*** *~20x~i5~300~ square inches.
(14.) 500 fathoms = (500 x 6 x 12) inches = 36000 inches ;
.*. if (2i be the density of atmospheric air and d^ be the density of
compressed air,
<t : di = (30 X 13-57) : (36000 x 1 -027) + (30 x 13-57)
= 407-1: 37379-1;
•••*-3ySi='^^^^^^^^y-
(15.) The water barometer under the same additional pressure
would rise (20 x 12) inches, or 240 inches ;
.-. the mercurial barometer will rise ,-;—-; inches, or inches.
13'57 ' 1357 '
or, 17j%Yr iiiches, or 1 foot 5j^^V inches.
(16.) When the floating body is partially immersed, both air and
water are displaced ; but the absolute weight of floating body = weight
of displaced fluids, which must therefore be constant. Therefore when
the barometer rises, there must be a smaller water displacement^ that
is, the body rises ; while a decrease in the atmospheric pressure, when
the barometer falls, will necessitate an increased water displacement,
and therefore the body will sink a little.
(17.) Pressure at sur£eu»=the atmospheric pressure.
Pressure at 15 inches below the sarface of mercury »
(atmospheric pressure at surface of water)
+ (pressure at depth of 17 feet of water)
+ (pressure at depth of 15 inches of mercury)
BB(atmo6i^eric pressure) +}> (atmospheric pressure)
+ 1 (atmospheric pressure)
■stwioe atmospheric pressure ;
.'. the pressures are as 1 : 2.
94 KE Y TO ELEMENTAR V HYDRO STA TICS.
(18.) A balloon is a hollow envelope, niade of some light material,
as silk, which, when filled with heated air, or some gas, rises in the
air. This must be the result, if the weight of the air displaced by the
balloon exceeds the weight of the balloon.
(19.) The external pressure on the bladder, increasing as the bladder
is forced lower into the water, compresses the bladder, till at last the
volume occupied by the air in the bladder is less than a volume of
water equal to the weight of the bladder and air contained therein,
and then the bladder must sink.
(20.) Let X be the height of the column in inches.
Then pressure on a square inch
= weight of X cubic inches of mercury
= weight of 13*56ic cubic inches of water
13-56a; x 1000
1728
13-56arxl000
1728
ounces.
= 226;
226 X 144 32544 „^ ^ . ,
••^=^1130 ^-ll3()-=28-^"^^^^-
• •
(21.) Area of piston =36 square inches.
Original pressure on piston =(36 x 15) lbs. = 540 lbs.
Then since the pressure of air varies inversely as the space it
occupies,
new upward pressure on piston : 540= 216 cubic inches : 72 cubic inches
= 3:1;
.*. upward pressure on piston =1620 lbs,
and downward pressure on piston by atmosphere =540 lbs. ;
.'. a weight of 1080 lbs. must be placed on it.
(22.) Height of water barometer, when the pump ceases to work,
is 364 inches ;
.*. height of mercurial barometer =--— inches = «/» inches
13*6 136
B 26^ inches.
KE V TO ELEMENTAR V HYDRO STA TICS. 95
(23.) Height of column of mercury within the bottle = twice the
height of mercurial column exposed to the ordinary pressure of the
atmosphere = 60 inches =5 feet.
(24.) Let X and y be the capacities of the receiver and the barrel.
Then^^±^=|-;
X 2
.*. 2x + 6y — 3x, and .*. x :y = 6 : 1.
(25.) The air will be compressed inside, and so will displace less
water ; and since the cylinder floated before the increase of atmos-
pheric pressure, it will now sink, because the weight of fluid displaced
is now less than the weight of the cylinder.
(26.) Let X and y be the capacities of the receiver and the cylinder.
Pressure after twenty strokes =^-i—^- (original pressure)
X
by
= five times original pressure.
(27.) Let X be the depth in feet
Then 31 :30 + a;=l :2,
or, 62 = 30 + a;, and .'. 05=32 feet.
(28.) Area of base of cube = 100 square inches,
and pressure on base = n 000 x jwao) o^Jices ;
. . pressure on a square inch of ba3e= (lO x ]^^ ounces
= 5j®o\ ounces.
(29.) Pressure on 16 square inches of surface = (236 x 16) ounces
= 3776 ounces.
Weight of column of water of (16 x 9 x 12) cubic inches
= ( 16 X 9 X 12 X _- j ounces= 1000 ounces ;
.*. whole pressure =4776 ounces.
96 KEY TO ELEMENTAR V HYDROSTATICS.
(30.) Let D be a point in the surface of the fluid, and DA
vertical line.
n „ T^rAoan ra of 7? DR
a
— c
-B
FiQ. 192.
rp, pressure at .B _ DB ^
pressure at A DA '
. pressure at J. — pressu re at B__DA — DB
pressure at -4 E2
Sunilarly,
_AB
DA
Sunilarly,
pressure at ^ - press ure at G __ DA - DC A G
pressure at A DA
By division,
DA'
ivision,
pressure at J. - p re ssure at B _ AB ^
pressure at JL — pressure at C7 AC*
p AB
9.
B
P
Pro. 193.
AC
p AB AB
"q-^p^AC-AB BC
(31.) An outer barrel ACDB encloses an inner barrel
OR into which the bullet is rammed.
Air is injected through the end of the outer barrel
through the stock P. When a quantity of air has been
confined in the barrel ACDB, a trigger at JK" is pulled
which opens the bottom of the inner barrel, and the air
being suddenly admitted projects the bullet from B with
^ much force.
r\
Q
(32.) iSfuppose the column of mercury
to rise from P to Q in the tube.
Then it sinks in the cup to 0, a point
such that 00=-! of OP.
17 ^
Hence the true variation is
Pg + lofPftor,J|ofPg.
Hence in graduating the scale the distances
actually measured from the zero point must be less
than the space indicated by the numbers placed
a^inst the graduations in the ratio of 17 : 18.
\U
C
Fxo. 104.
KE V TO ELEMENTAR V HYDROS TA TICS: 97
(33.) Pressure of air on a square inch
= weight of a column of mercury resting on a square-inch base
= weight of 30*5 cubic inches of mercury
=weight of (30*5 x 13*57) cubic inches of water
= 30*5 X 13*57 X 252*6 grains
= 104547*351 grains.
Now 7000 grains troy = a pound avoirdupois, and .*. 104547*351
grains =14*935 lbs. nearly.
(34.) The pressure of air in the upper part of the tube, when the
instrument stands at 27 inches, is equal to the weight of 3a cubic
inches of mercury, a being the area of the base of the column in the
tube.
The air occupies a space of 9a cubic inches.
Hence if this air be subjected to a pressure of 30a cubic inches of
mercury it will be reduced to atmospheric density, and if a; be the
space it then occupies in cubic inches,
xi9a=2a : 30a ; (Hydrostatics, Art. 80.)
••^"30 ""10'
.*, the air will occupy ^q of an inch in length.
(35.) Actual weight of stones + weight of air displaced by stones
= actual weight of the weights + weight of air displaced by weights.
Now the weights are smaller in volume than the stones.
.*. weight of air displaced by stones is greater than weight of air
displaced by weights.
And a diminution of atmospheric pressure will decrease the weight
of air displaced in each case in proportion to the weight of air 'displaced
before in each case.
.*. it will decrease the weight of air displaced by stones more than
it decreases the weight of air displaced by weights.
.'. the stones will appear to weigh more than their actual weight ;
.'. he will gain by selling when the barometer is low.
O
98 KEY TO ELEMENTARY HYDROSTATICS.
(36.) The confined air is pressed by a column of mercury equal to \
atmospheric pressure^ and by the atmospheric pressure at the sur&ce
of this mercury.
.'. we have a body of air at atmospheric density subjected to a
pressure | times the atmospheric pressure ;
.*. the air will be compressed into § of the space it occupied before ;
•*. the mercury descends } of 14 inches, or 4J inches.
Examples — VI. (p. 75.)
(1.) It will increase the time of fiUing the receiver, because the only
effecti/oe work will be done by the descending piston after it has passed
the hole. Hence, if the aperture be made one-third of the way up the
barrel, the distance through which the piston acts effectively being
only one-third of the distance through which it worked effectively at
first, three times the original time will be required for filling the tank.
(2.) The tension of the piston-rod = pressure of atmosphere above
— pressure of atmosphere below.
Now pressure of atmosphere above = (4 x 15) lbs. = 60 lbs., pressure
of atmosphere below = (4 x 15) lbs. — weight of a column of water 16
feet high resting on a base of 4 square inches
= 60 lbs. - weight of n 6 x - 7, ) cubic feet of water ;
.'. tension of rod = weight of — cubic feet of water.
•7
iaaa\ 4x1000
1000) ounces =
= (|xl000)
=^|? lbs. =27 J lbs.
9 X 16 lbs.
(3.) (a) K the hole be made in the longer arm, hehw the level of
the shorter arm, no effect will be produced.
(P) If the hole be made in the longer arm, above the level of the
shorter arm, all the fluid in the longer arm below the hole wiU
descend, and all above in the same branch will ascend, causing the
remainder of the fluid to flow through the short branch, till the siphon
is emptied.
KE V TO ELEMENTAR Y HYDR OSTA TICS. 99
(y) If the hole be made in the shorter arm, all the fluid below the
hole in this arm will descend ; all above in this arm will ascend and
flow through thfe longer arm, emptying the siphon.
(d) If the hole be made at the top of the siphon, the fluid will
descend in each arm, and will empty the siphon.
(4.) A height equal to the height of the water barometer at the time,
which is (13-57 x 29) inches, or 393*53 inches, that is, 32 feet 9*63
inches, or 32*79416 feet.
(5.) The fluid would descend in each branch, and the siphon would
be emptied.
(6.) Equally well at both.
(7.) No ; because the water in the hold must be below the level of
the surface of the water in the harbour.
(8.) The height of the water barometer at the time, and this is
(taking the specific gravity of mercury as 13*57), (30 x 13'57) inches,
or 407'1 inches, that is, 33 feet 11*1 inches.
(9.) If the air be removed from the siphon, the fluids will first
ascend in each arm, and then the flow from the longer arm will
commence and go on in the usual manner.
(10.) The water will rise in the inverted tube as high as the top of
the inserted tube, and then it will flow out of this tube.
(11.) The water would soon cease to flow, because there would be
no atmospheric pressure to cause a continuous flow. But when the
air is gradually re-admitted, the water will begin to rise in each
branch, and ultimately the flow will go on regularly.
(12.) (a) The water will flow into the lower vessel.
(6) The water will descend in each arm of the siphon till it
stands at a height above each surface equal to the height
of the water barometer,
(c) The water will flow into the lower vessel
(13.) The shorter arm must be 2 feet in length, and the longer arm
just more than 2 feet.
I
loo J^EY TO ELEMENTAR Y HYDR OSTA TICS.
Examples— VII. (p. 8i.)
(1 .) 0= I {F- 32), and U = i.(J' - 32) ;
.•.(l)C=|-(30-32)=^=-lJ.
JJ=-i(30-32)=-|.
.(2)0=-|(45-32)=|-xl3=7f.
ie=i.(45-32)=-|-xl3=5f
(3) (7=|-(56-32)=-| X 24 = 13J.
i{=i.(56-32) = -| X 24=10§.
(4)C=|(0-32)=^=-17f
9 ^" " ' 9
|(0-32)=-4?
B=4(0-32) ^=-14J.
(5)C=|<-7-32) ^=-21§.
JJ=-i(-7-32)=^=-17J.
(6) C=|(-45-32) = ^= -42$.
iJ=|(-45-32) = -|??=-34S.
(2.)C=5j,andJ'=^+32;
/. (1) C=^=f =6i.
^^=^+-32 = 11^ + 32=431,
• KEY TO ELEMENTARY HYDROSTATICS. loi
(3)C=°J«xO.
^^=0 + 32=32.
(4)0=^=-22J.
^=^15? + 32=-40j + 32=-8}.
(5)C=5iii^ = 5x(-16) 80.
^=— ^~^'^' + 32 = (9x -16) + 32=-112.
(6)0=^^=150.
J'= ?-^i^ + 32 = 270 + 32 = 302.
(3) ^^=-1- 0+32, and 1?=^;
O 5
q y 1ft
.-.(1) ^=--g-+32 = 28J + 32 = 60jt.
P_ 4xl6 _64
(2) ^=?4^ + 32=81 + 32=113,
D
T> 4x45 . _ „^
iJ=—_ --=4x9=36.
(3)jp=^i^ + 32 = 198 + 32=230.
jR=i^^=4x 22=88.
102 KEY TO ELEMENTARY HYDROSTATICS.
(4)J'=?^+32=32.
o
(6) j'=?iLtil) + 32= -27+32-6.
28-121^= -12.
o
(4.) TeSy if the graduations are to be onifomi.
(5.) 9C=5 (J'-32) ) .
0+ ^^=60 J '
.-.90=5 (60-0-32),
or, 90=300-50-160;
/. 140=140, and /. 0=10, and ^=6a
(6.) 90=5(^-32) )
.-.90=5(50-32);
/. 160=160, and /. 0=10, and J?'=60.
(7.) 90=5 (J*- 32))
.-.90=5(0-32);
.-. 40= - 160, and .'. 0= -40, and jP= -40.
(8.) (80- 20)% or 60" on the new scale = (80- 32)', or 48"-P;
.-. each degree on the new scale =--^ J*, or — ^-
(9.)C=-|(-P'-32)
9
^
.-. 0-1 (78-3!)-?-^-^-i!5|.
KEY TO ELEMENTAR Y HYDROSTATICS. 103
(10.) 90= 5 (J?*- 32)) .
(7+J'=88 J '
.-.9(7=5(88-0-32);
.-. 140=280, and .-. 0=20, and ^=6a
(11.) 0-|-(J'-32);
.•.0=|- (49-32);
••^-"9 9" ^*
(12.) The graduations would be inconyenientlj smalL
(la) 90=5 (J*- 32))
J>'=30 5 '
.-.90=5(30-32);
.-. 6O-160, and .'. 0=265, "^d J'=8a
(14) 9O=5F-160) .
170= 5 J* S '
.-. 90=170-160, and .•. 0=20, and J?'=6a
(15.) 90= 51^-160)
.-. 90= - 50- 160, and .'. 0- - llj, and J^=llf.
(16.) Let X be the number of degrees on the latter.
rp, 15-9 a;-12
^^^^i5:To=x-ri4>
.-. 6(a;-14)=5(a;-12);
.-. 6a;-84=5x-60, /. «=24
(17.) Let X be the number of degrees on the latter.
16-8 x-11
Then
16-10 x + 14'
.-. 8(x-14)=6(x-.ll);
/. 8x-112=6x-66, and.-. «— 23.
1
1 04 KEY TO ELEMENTAR Y HYDRO ST A TICS.
aa)ii=-|(J'-32)
\
.•.922=4(22 + 47-32);
.-. 522=60, and .-. 22=12, and J?'=59.
Also, if (2 be the number of degrees in the difference,
22=4 (J?*-
22=
^=|(^-32))
;=J'-47-d )
.-. 4P- 128=9 (J?'-47-d), or, 5^=295 + 9(2, or, J'=59 + 4-(i;
5
.'. F riaes ?? and .*. 22 rises ~
Examples — ^VIII. (p. 87.)
1. Let X be the yolume of the part immersed, y the volume of the
part out of the water.
m, a: -925
Then = , ^,^ ;
x + y 1-026'
.-. 1025a; = 925a5 + 925y ;
.-. 100a;=925i/;
and ,\ y\x=\QO: 925=4 : 37.
2. Let X be the volume of the body, y the volume of the part out
of the fluid in the first case, s the specific gravity of the body.
Then — -^-tt,
X -9'
and^=.-T ;
■ X 1*1 *
. x-y_V\^
•• y --9 '
. * 1-11 1. ??
"y '"9'^^'y = 9-
^^°^20'n'
and.-.«=|^=-495.
KE Y TO ELEMENTAR V HYDROSTA TICS. 105
3. f=^+32;
5
.-. whenC=-40%
J^= -72 + 32= -40;
and when 0=350%
^^=630 + 32=662.
4. After three strokes
(3 \'
- — -J . original density ;
27
.*. pressure =-— of original pressure ;
27
.*. height of barometer =_- of 28 inche3=ll*8125 inches.
u4
5. Let Wi and w^ be the weights of the hydrometers, v the volume
of fluid displaced in each case, Si and ^ the specific gravity of the
fluids.
Thenii=.^>=-^=1.
6. Let V be the volume of the man, s his specific gravity ;
/. :^ is the volume of his head.
\^=volu.eofmanunMe«ed.
— = volume of bladders :
12
.*. volume of water displaced =——.
Then v . « = -^^ . 1, or, « = {|= I'OSS.
7. Height=(28 x 13*6) inches=380-8 inches=31 feet 8-8 inches.
8. C-f- (i?'- 32) =1 (27-32)= -^=-2|.
10(5 KEY TO ELEMENTARY HYDROSTATICS.
I
9. Let X be the depth in feet reached by his fingers,
15 .
jc - — is the depth reached by his feet ;
15
•*. a; : a;- — =3 :2^
or, 2a;=3a5- 22J, and .*. jc=22j feet
10. Let X be the size of the cork in cubic feet.
Then weight? of cork=240a5 ounces, and weight of mercury dis-
placed =13600x ounces ;
.-.« (13600 -240) = 167;
^ 167 ^ 1
•'•^ 13360 80*
11. Let X be the number of degrees on De Lisle.
150-x 47-32
Then
or,
150 180 '
150 -a 15
> R « >
5 6
.•. 900 - 6j5=75, or, 6x=825 ;
.-.a; =137 J.
12. Let «= volume of cork.
Then since ('0013 x n)= specific gravity of n atmospheres,
vx-24=wt;x-0013;
•24 2400
.*. n=
•0013 13
13. The body will rest when reduced to ^Z/ its natural size ;
20 + n JL^
**• 20 + 471*" 2 '
.'. 40+2n=A20 + 4n;
.'. n=10 feet,
14 (a) When it is at depth of 5 feet,
20 + n _ 20 + 5 _ 6 ,
20 + 4n~20 + 20~ 8 '
.•. body displaces — of 20 lbs. of water, or, 12 J lbs. of water ;
8
.•. since the body weighs 10 lbs., we must have a downward force
of 2 J lbs.
KEY TO ELEMENTARY HYDROSTATICS. 107
03) When it is at depth of 30 feet,
20 + n _ 20 + 30 _ 50 _^ ^
20 + 4n""20 + 120""l40""l4'
.•. it displaces — of 20 lbs. of water, or, 1\ lbs. of water,
.*. we must hare an upward force of 2f lbs.
16. When the body displaces 11 lbs. and 9 lbs. of water respectively.
20 + n 11
(1) When
(2) When
20 + 4n 20'
or, 400 + 20n « 220 + 44n,
or, 24n=180, and .*. n=7J feet.
20 + w 9
20 + An 20'
or, 400 + 20w = 180 + 36n,
or, 16n=220, and .-. n=13| feet
16. Let F be the reading on the Fahrenheit scale.
Then -^ (^-32) is the reading on Reaumur's scale.
•7
.-. JP:i- (^-32)=25:4;
.-.36^=100^-3200.
Hence ^^=50.
17. Let F be the reading on the Fahrenheit scale.
Then -^ (-^- 32) is the reading on Reaumur's scale,
a
and -^ (#-32) is the reading on the Centigrade scale.
.-. i^+ -| (J* - 32) + 1. (-F- 32) = 212 ;
.% #+F-32=212;
.•.J'=12
I o8 KEY TO ELEMENTAR V HYDR OSTA TICS.
18. Let w be the apparent weight of each in water.
«
(Hydrostatics, Art. 65, Case I.)
rru 2-6 22
^^^^ T = 223^'
,7-8 n
and-r-= ;
1 n-w
99
2*6 '
an(i=-s=»-M;
7*8
22 n _
/. 66-n=7-8 (22 -n) ;
.*. 6*871= 105*6, and .*. n=15^.
19. Let w be the apparent weight of the substances in the fluids,
s the specific gravity of the fluid.
Then jp,
.•.-g-=l-^,
and
38
2-25
= 3-1(7
/. -I — 4- = 2, or,4«-«=6, and /. « = 2.
•75 3
k
N
vv
20. Let N be the position of the fulcrum, v the volume of each
body.
C Then tension of string AB—v (2*7 - 1),
and tension of string CD=v (6*1 - 1) ;
.*. if CN=x inches,
i; (2-7-1 )^ X ^
V (6-1-1) 71 -a'
17 X
, or, 7l-a=3«;
6
B
6
D
Fio. 196. 51 71 -a
/. a!=-7- inches =17} inches.
4
KE Y TO ELEMENTAR Y HYDROSTATICS, 109
Examples — IX. (p. 94.)
1. Let X and y be the lengths of the aims at the ends of which the
gold and silver hang in each of the three cases : then
(l)a;:y=10-4:19-4
= 104:194
= 52:97.
(2) x: 2/= 10-4-1 :19'4-1
= 9-4: 18-4
=47 : 92.
(3) In this case the silver is forced up by the mercury,
and acts on the lever vertically upwards, hence
the fulcrum must be at one end, and the gold
between the silver and the fulcrum, and
X : t/= 13-5 - 10-4 : 19*4 - 13*5
= 3-1 :5-9
= 31:59.
2. Let X be the weight of the body in the fluid.
r^ 1 -X 3
Then-V^=-^;
7 7 '
.•. 05=4 lbs.
Then, by 8tcU.ic8^ Art. 102,
height : base =3 :4.
3. Let V be the volume of hydrogen in the balloon in cubic feet.
Then v x '07 x 1*3= weight of hydrogen in ounces,
and V X 1*3= weight of displaced air in ounces ;
.*. V X 1-3=1; X -07 X 1*3 + 10 X 112 X 16 ;
.'. V X 1-3 X -93=10 X 112 X 16 ;
4. Let pi denote the pressure internally before the ascent, p^ the
pressure internally after the ascent.
Then ^2 :i>i=3 :4.
Hence if Vi and Va be the volumes of the gas before and after the
ascent.
1 10 KEY TO ELEMENTAR Y HYDROSTA TICS.
Hence one-third of the gas must have been expelled to mftintj^in
equilibrium between the external and internal pressures.
Also, since the pressure of the external air is only three-fourths of
the pressure of the external air before the ascent, one-fourth of the
whole weight must have been thrown out.
5. The gas has been expelled to preserve equilibrium of internal
and external pressures, and the ballast to preserve equilibrium of
vertical pressures on the balloon.
6. Volume of cylindrical vessel =7r . 4^ . 9 cubic inches.
Volume of interior of cylinder =7r . 3^ . 8 cubic inches ;
.'. volume of wood =7r (16 x 9 - 9 x 8) cubic inches = 72 tt cubic inches,
also, volume of cylindrical mass of fluid displaced = tt . 4^ . 3 cubic in. ;
.'. if s be the specific gravity of the wood,
727r.s=487r;
_48_^
•'•*"72"3*
Again, let x oe the depth to which it sinks.
Then volume of fluid displaced =7r.42 + 7r (42-3^ {x-\)
= 77ra; + 97r;
. *. 77rx + 97r = 727r . s = 487r ;
.'. 7a5=39, or, a;=5f inches.
7. No change will occur till the stone falls from the ice ; but then
the ice will displace less water than before, and the surface of the
water will sink.
8. Let AB be the surface of the water outside, CD the level to
which the water rises inside.
p^ g Then, by Boyle's law,
~ pressure of air in ^Z) : atmospheric pressure = jB^ : BB,
Hence if h represent the atmospheric pressure, and x
D be the length of BD, and y the length of DN^
pressure of air in AB x + y ^
atmospheric pressure x '
M N ...?+^=^or,x2 + x^=x^ + %,and.-.a:2=^y^
Fia, 196 h X
KE Y TO ELEMENTAR Y HYDROSTATICS. 1 1 1
9. The pressure is equal to the weight of (80.x 6) cubic feet of water.
Hence if we take the weight of a cubic foot of sea water as 1000
ounces (which is rather less than the actual weight),
pressures (80 x 6 x 1000) ounces = 30000 lbs.
If the box were not water-tight the water would enter it, and the
pressure will be the same inside as outside.
10. Let X be the number of feet to which the bag is sunk.
The pressure on the bag will then be equal to a pressure indicated
by a height of (a: + 34) feet on the water barometer.
•''19 a;+34'
.'. ic= (18 X 34) feet = 102 fathoms.
11. Let a; be the weight of the vessel in ounces.
Then, since the weight of a cubic foot of water is 1000 ounces,
a; + 2- X 1000 =1000;
8
1000 ,-,-
.'. a5= - = 125 ounces,
o
12. The surface of the fluid will rise higher than the surface of the
water, because the specific gravity of the fluid is less than the specific
gravity of the water.
13. Let % be the depth in inches to which the cylinder sinks.
Then 15 x I'l + 15 x -25=0; x 1 ;
.-. a;=15 X 1-35 = 20*25 inches.
14. The larger piece of cork will rise to the surface, and the length
of the string between it and the pulley will be 2 feet, while the
smaller piece of cork, at the end of 1 foot of string from the pulley,
will be entirely submerged.
Then if w and Zw be the weights of the pieces of cork, the smaller
piece displaces water to the weight of 4t(;.
Hence we must have an upward force =:3u; at end of longer string.
.'. the larger piece of cork must displace water to the weight of 6w,
end .'. it must be half immersed.
\
112 KEY TO ELEMENT A R Y HYDR OSTA TICS.
15. As the air is drawn away from the tube the water will rise in
both branches, and if the height of the top of the tube above the
surface of the water in the reservoirs is not too great, there will
ultimately be a continuous column of water in both parts of the
siphon, and then a regular flow will commence
I
16. Let V be the volume of each body, s^ and «2 the specific gravities
of the bodies, p^ and p2 the specific gravities of the liquids.
Then, «i-pi=«2~P2> o^> *i~*2=Pi~"P2> (1)
And«i-^ = S2-Pi, or, «i-S2 = ^^-2-^ • ...... (2)
Hence pi-p2 = ^^^ ^\
and thus pi = -^, and 8^ - gg = - ^*
Let X be the part of the heavier body immersed in the final experi-
ment.
Then vs^^^vs^-oyvl^ — ^j ;
X
.-. Si-S2=--o--0'i + p2);
. _P2 ^ . 7p2
•• 4" 2 4'
and thus « = -=-•
17. ^Z> is the level of the mercury, B is the height of the water.
Then the question to resolve is whether the weight of a column
of mercury in height CD in the longer
arm is greater (or not) than the weight
of a column of water, height CRy on the
same base, since the pressure upwards
'^at the bottom of this tube = pressure
- D upwards at ^ + weight of such a column
C of water ;
Fio, 197. .*. mercury will flow through the tube
if CD X density of mercury be greater than CR x density of water,
or, if zp be greater than /p',
or, —^ be greater than — •
KE V TO ELEMENTAR Y HYDR OSTA TICS. 1 1 3
18. Let a; = depth of air at the top of the vessel when the water
stands at the top of the pipe.
The amount of water forced in at each
stroke \b Al.
Then, if tt be the atmospheric pressure,
and p the atmospheric density,
pressure of confined air
= — 7r — (Ji- {c — x) )p + 7r, and 7r = ph ;
X
.*. — = 2/i-c + x,
X
and .'. x=
2h±\f4h^+"c^
X
l«
Pio. 198.
2
Again, considering the quantity of water
in the air-vessel and pipe, after n strokes
of the piston.
Aim— {c-x) A+ a(h -c + x) ;
2AI
19. Let x be the weight of the cylinder in lbs.
Thenaj + 6:x=8 :5;
.*. 5x + 30 = 8x, or, a; =10 lbs.
20. Let «! and 53 be the specific gravities of the metals, v and w the
measures of the volumes and weights mixed.
Then vsj^ + vs^ = 2v . 9,
'2w
8f ^
, It? w
and — H —
8,
^1 "t" ^2 — lOy.
1 + 1 = 1
«i «2 40'
and hence we find s^ = 10 or 8, aiid «« = 8 or 10.
21. Let X be the height of the cylinder.
Then, since the air which stood at height x inches under the
atmospheric pressure is condensed to the height of 1 inch,
1 :a;=15 : 165 ;
. •. .T»= -TT = 11 inches.
lf>
H
i
114 KEY TO ELEMENTAL Y HYDRO STA TICS.
2 2
22. The density of the air in the receiver of the air»pump is —- of -5-,
3 3
or -^ of what it was at first.
•7
The density of the air in the receiver of the condenser is after the
3
first stroke — of what it was at first.
2
The air in the barrel at the commencement of the second ascent of
the piston is — of the density of atmospheric pressure.
When the piston has reached a point — of the length of the barrel
from the valve of the condenser, the air in the barrel will be of tho
same density as the air in the receiver of the condenser.
The valve will then open, and the air will enter.
The air which occupied a space ( 1 + -3^ ) o^ t^e receiver will now
occupy a space equal to the receiver ;
.*. density will be — of the density of air in receiver ;
«7
11 3 33
.•. density will be — of — , or — of original density ;
.*. pressure in receiver of condenser : pressure in receiver of air-pump
=?? . j4
18* 9
:-33:a
£DfnImrgf) Snibereiti! ^retts:
THOMAS AKD ARCHIBALD CONSTABLE, PRINTERS TO R<R MAJBSnr.
Eibfngton'jS JEatfjematical Stxits
By J. HAMBLIN SMITH, M.A.
of Gonville and Caius College^ and late Lecturer at St, Peter's
College^ Cambridge,
A TREATISE ON ARITHMETIC. 35. 6d,
A KEY TO ARITHMETIC. 9s,
ELEMENTAR Y ALGEBRA, Part I, 35.
Without Answers^ 2s. 6d.
A KEY TO ELEMENTARY ALGEBRA, ^s.
EXERCISES ON ALGEBRA. 2s, 6d.
(Copies may be had without the Answers.)
ELEMENTS OF GEOMETRY. Containing Books
I to 6, and portions of Books 1 1 and 12 of Euclid^ with Exercises
and NoteSj arranged with the Abbreviations admitted in the
Cambridge Examinations. 3J. dd,
PART I, Containing Books I and 2 of Euclid^ limp cloth^ is. 6d.
may be had separately.
KEY TO ELEMENTS OF GEOMETRY, Zs. 6d,
ELEMENTAR Y STA TICS, y.
ELEMENTAR Y HYDROSTA TICS. 35.
A KEY TO ELEMENTARY STATICS AND
HYDROSTATICS, ds.
ELEMENTARY TRIGONOMETRY, 45. M.
KEY TO TRIGONOMETRY, ^s, 6d.
BOOK OF ENUNCIATIONS, For Geometry,
Algebra J Trigonometry y Statics j and Hydrostatics, is,
AN INTRODUCTION TO THE STUDY OF
HE A T. 3 J.
By E. J. GROSS, M.A.
Fellow of Gonville and Caius College, Cambridge, and Secretary to
the Oxford and Cambridge Schools Examination Board,
ALGEBRA, Part IL Zs. 6d.
KINEMATICS AND KINETICS. 5^. 6d.
By G. RICHARDSON, M.A.
Assistant Master at Winchester College, and late Fellow of
St. John^s College, Cambridge.
GEOMETRICAL CONIC SECTIONS. 4X. dd.
BY THE SAME AUTHOR-
Crown 8vo, jj. 6d.
Exercises on the Elementary Principles
of Latin Prose Composition.
With Examination Papers on the Elementary Facts of Latin Acci-
dence and Syntax.
^ Key, Crown 8vo, 5^.
Crown 8vo, 3.r. dd.
An Elementary Latin Grammar.
Small Svo, 2J. 6</.
The Rudiments of English Grammar and
Composition.
Crown Svo, 4^. dd.
An Elementary Greek Grammar.
Crown Svo, 25. 6d.
Short Notes on the Greek Text of the
Gospel of St. Mark.
Crown Svo, \s, 6d.
Notes on the Greek Text of the Acts of
the Apostles.
3, Waterloo Place, Pall Mall,
October^ 1882.
€iiucattonal W&it\t^^
published by
Messrs. RIVINGTON
New Books in Preparation
AND IN THE Press.
Studies in Philosophy, Ancient and Modern. By w.
L. Courtney, M.i\,, Fellcnv and Tutor of New College, Oxford.
Contents. — Ancient Idealism, Parmenides — Ancient Hedonism, Epicurus — The
Failure of Berkeley's Idealism — A chapter in the history of the word "Cause" — ITia
New Psychology — The New Ethics — Back to Kant"— Kant as a Moralist and as a
Ix>gician — The Hegelian Religion. Zvo. 12s. [No7V ready,
Etyma Graeca. An Etymological Lexicon of Classical Greek.
By E. R. Wharton, M.A., Lecturer and late fellow of Jesus
College, Oxford. Cro7vn Svo. js. 6d. [Alow ready.
A Geography, Physical, Political, and Descriptive,
for Beginners. By L. B. Lang. Edited by the Rev. M.
Creighton, M.A., late Fellow and Tutor of Merton College,
Oxford. With Maps. Small %vo.
Vol. XL— the CONTINENT OF EUROPE. \_Now ready.
Vol. IIL— ASIA, AFRICA, AND AMERICA. [In preparaHon.
EXCerpta Facilia. a Second Latin Translation Book, containing
a Collection of Stories from various Latin Authors, with Notes at
end, and a Vocabulary. By H. R. Heatley, M.A., and H. N.
KiNGDON, B.A., Assistant- Masters at Hillbrow School, Rugby,
Crffion %vo. 2s, 6d. [Now ready,
A Key for the use of Tutors only. $s.
WATERLOO PLACE, LONDON.
MESSRS. RIVINGTON'S
New Books in Preparation and in the Press — tmtvnued,
A Handbook in Outline of the Political History of
England to 1881. Chronologically arranged By Arthur H.
D. ACLAND, M.A., Christ Churchy Oxford^ and Cyril Ransome,
M.A., Prcfessor of Modem Literature and History, Yorkshire
College, Leeds, Crown ^o. 6s. [Lately published.
Stories from English History. By Louise Creighton,
author of ** A First History of England," *'Life of the Black Prince,"
&c. Royal i6mo. Illustrated, [J^^t ready.
Introduction to Greek Verse Composition. ^;/ Arthur
SiDGWiCK, M.A., Tutor of Corpus Christi College, Oxford; late
Assistant' Master at Rugby School, and Fellow of Trinity College,
Cambridge; andY, D. Morice, M.A., Assistant- Master at Rugby
School, and Fellow of Queen^s College, Oxford. [In preparation,
CliVUS. Elementary Exercises in Latin Elegiac Verse. By A. C.
Ainger, M.A., Assistant-Master at Eton College, Craivn %vo.
Part I. 2s, 6d. Part II. 2s. 6d.
A Key for the use of Tutors only. y. 6d. [J^st published.
A Syntax of Attic Greek for the use of Students
and Schools. Part I. The Simple Sentence. Part II. The
Compound Sentence. Part III. Prepositions, Particles, and
Figures. ByF. E. Thompson, 'M. A. ^ Assistant-Master at Marl-
borough College. Crown Svo. [In the press.
A Second Latin Reading Book. Forming a continuation of
Easy Latin Stories. By G. L. Bennett, M.A,, Head-Master of
the High Sihool, Plymouth, Crown Svo, 2s. 6d. [A^ow ready,
A Key for the use of Tutors only. 5j-.
Manuals of Religious Instruction. Edited by John
Pii.kington Norris, D.T>,, Archdeacon of Bristol, and Examining
Chaplain to the Bishop of Manchester. The Old Testament. The
New Testament. The Prayer Book. New Editions, partly re-toritten.
Small %vo. y. 6d. each. [In the press.
(The Original Editions may still be had.)
%
WATERLOO PLACE, LONDON.
EDUCATIONAL LIST,
New Books in Preparation and in the Presa— ^ofi^mfiMt.
A Manual of Greek Verbs, with Rules for the Formation of
Tenses, and Tables of Verbs for Practise. By F. Ritchie, M. A.,
and E. H. Moore, M.A., Assistant- Masters in the High School^
Plymouth. Crmim Svo, [Now ready.
A Companion to Algebra. By Leonard Marshall, M.A.,
Assistant- Master at Charterhouse School. Crown %vo. [In the press.
The Jugurtha of Sallust. By E. p. Brooke, M.A., Assistant-
Master at Rugby School. [In preparation.
Bacon's Essays. Complete Edition. Edited by ¥. Storr, M.A.,
Chief-Master of Modern Subjects in Merchant Taylors' School^ and
late Assistant-Master in Marlborough College. [In preparation.
Essays on Aristotle. By Various Writers. Edited by Evelyn
Abbott, M. A., LL.D., Fellorw and Tutor of Balliol College^ Oxford,
[In preparation.
Introduction to the Study of Greek Antiquities.
Edited by J. S. Reid, LL. M. , Fellow and Assistant- Tutor of Gonville
and Caius College^ Cambridge; Classical Examiner in the University
of London. [In the press.
History of the Romans to the Establishment of
Iraperialism. By]. S. Reid, LL.M. [In preparation.
Selections from ThUCydideS. An Easy Greek Reading Book.
By E. H. Moore, M.A., Assistant-Master at the High School^
Plymouth. [In preparation.
Arnold's First Greek Book. By Francis David Morice,
M.A., Assistant- Master at Rugby School^ and Fellow of Queen* s
College, Oxford. A New and Revised Edition. Crown %vo,
[In the press.
A Short History of England for Schools. By f. York-
PowELL, M.A., Lecturer at Christ Churchy Oxford. With Maps
and Illustrations. Small Svo. [/n the press.
A Latin-English Dictionary for Junior Forms of
Schools. By C. G. Gepp, M.A. i6mo. [In the press.
WATERLOO PLACE, LONDON.
MESSHS. RIVINGTON'S [BNOLISH.
ENGLISH SCHOOL-CLASSICS
With Introductions, and Notes at the end of each Book.
Edited by FRANCIS STORR, M.A.,
CHIEF MASTER OF MODERN SUBJECTS AT MERCHANT TAVLORS' SCHOOL.
Small 8z/<7.
Thomson's Seasons : Winter. WUh introduction to the Series,
by the Rev. J. Franc K Bright, M.A., Master of University
Collie, Oxford, is,
Cowper'S Task. By Francis Storr, M.A. 2s.
Books I. and II., gd. ; Books III. and IV., gd. ; Books V. and VI., gd,
Cowper'S Simple Poems, with Life of the Author.
By Francis Storr, M.A is.
Scott's Lay of the Last Minstrel. By]. Surtees Phill-
POTTS, M.A., Head' Master of Bedford School. 2s. 6d.
Canto I., gd. ; Cantos II. and III. gd. ; Cantos IV. and V., gd. ; Canto VI., gd.
Scott's Lady of the Lake. By R. W. Taylor, M.A., Head'
Master of Kelly College, Tavistock. 2s.
Cantos I. and II., gd. ; Cantos III. and IV., gd. ; Cantos V. and VI., gd.
Notes to Scott's Waverley. By H. W. Eve, M.A, Head-
Master of University College School, London, is,; or with the Text,
2s. dd.
Twenty of Bacon's Essays. By Francis Storr, m.a. is.
Simple Poems. Edited by W. E. Mullins, M.A., Assistant-
Master at Marlborough College. &/.
Selections from Wordsworth's Poems. By H. h.
Turner, B. A, late Scholar of Trinity College, Cambridge, is.
WATERLOO PLACE, LONDON,
ENGLISH.] EDUCATIONAL LIST.
Wordsworth's Excursion : The Wanderer. By H. H.
Turner, B.A. is.
Milton's Paradise Lost. By Francis Storr, M.A. Book I.,
9^. Book II., 9^.
Milton's L' Allegro, II Penseroso, and Lycidas. By
Edward Storr, M.A., late Scholar of New College^ Oxford, is.
Selections from the Spectator. By Osmond Airy, M.A.,
Assistant-Master at Wellington College, is,
Browne's ReligiO Medici. By W. p. smith, M.A., Assistant-
Master at Winchester College, is.
Goldsmith's Traveller and Deserted Village. By c.
Sankey, M.A., Head' Master of Bury St. Edmund* s Grammar
School, is.
Extracts from GrOldsmith's Vicar of WakelQleld. By
C. Sankey, M.A. is.
Poems selected from the Works of Robert Burns.
By A. M. Bell, M. A., Balliol College^ Oxford. 2s,
Macaulay's Essays.
MOORFS LIFE OF BYRON. By Francis Storr, M.A. gd,
BOSWELL'S LIFE OF JOHNSON. By Francis Storr,
M.A. gd.
HALLAM'S CONSTITUTIONAL HISTORY. By H. F.
Boyd, IcUe Scholar of Brasenose College^ Oxford, is,
Southey's Life of Nelson. By w. E. Mullins, m.a.
2s. 6d,
Gray's Poems. Selection from Letters, with Life
by Johnson. By Francis Storr, M.A. is,
*^* The General Introduction to the Series will be found in Thomson's Wintbk.
WATERLOO PLACE, LONDON
A
8 MESSJiS. RIVINGTON'S [HISTOBT.
HISTORY
With numerous Maps and Plans, New Edition^ Revised, Crown %vo,
A History of England. By the Rev. j. franck bright, m. a.,
Master of University College^ and Historical Lecturer at Balliol, New,
and University Colleges^ Oxford; late Master of the Modem School
at Marlborough College.
Period L— MEDIiEVAL MONARCHY : The departure of
the Romans, to Richard III. From A.D. 449 to A.D. 1485.
4r. 6d.
Period II.— PERSONAL MONARCHY : Henry VII. to
James II. From a.d. 1485 to a.d. 1688. 5^.
Period III.— CONSTITUTIONAL MONARCHY: WUliam
and Mary, to the present time. Fro^l A.D. 1689 to A.D. 1837,
^s. 6d,
Extract from the Regulations for the Army Examinations,
" At the competitions for the Military Colleee, Sandhurst, the Academy, Woolwich,
&c., the examinations in English History will be limited to the periods a.d. X760-1790,
and 1790-1820.
" %* The candidate reading on the period selected should include the study of that
part of Bright's History which treats of the period he selects."
With Maps and Illustrations, Small Svo,
A Short History of England for Schools. By
F. York- Powell, M.A., Lecturer at Christ Church, Oxford,
\In the press.
With Forty Illustrations, i6mo. 2s, 6d,
A First History of England. By louise creighton.
Author of ''Life of the Black Pnnce," "Sir Walter Ralegh," <Sr*r.
"This is a neat, well-written little volume of about four hundred paees, in which the
whole story of English history is told in a succinct but interesting way. — Athenaeum.
" 'I'his is one of the most satisfactory of the many histories of England lately published
fjr schools. Mrs. Creighton has spjared no pains to make their earliest lessons in English
history in every way attractive to little children."— ./cYi^^w/y.
"This is really a charming little volume." — yournal of Education.
"The book gives a pleasing summary of the more striking events, and is written in
simple and familiar language. Forty illustrations (which we are told are from 'authentic
sources ') brighten up the narrative and attract by their occasional quaintness. The book
contains 383 well-printed pages, and there is a carefully prepared index." — Schoohnaster.
"The style is so simple, tnat the book cannot fail to be understood by any child who
can read ; and at the same time, the crucial and decisive events of our history are treated
with an adequate appreciation of their importance, and their causes and consequences
clearly set forth. The interest of the book is increased by numerous illustrations, many
of which are reproductions of old ditsigns" —-Scotsman.
" It is well and carefully written, and the illustrations are numerous and authentic"
School Guardian.
WATERLOO PLACE, LONDON.
HISTO&Y.] EDUCATIONAL LIST.
Historical Handbooks. Edued by oscar browning,
M. A., Fellow of Kin^s College^ Cambridge,
Crown &/£?.
English History in the XIYth Century. By Charles
H. Pearson, M.A., late Fellow of Oriel Collie, Oxford, 3j. 6d,
The Reign of Lei^iS XI. By P. F. Willert, M.A., Fellow of
Exeter College^ Oxford, With Map. y. ^
The Roman Empire. A.D. 395-800. By A. m. curteis,
M.A. With Maps. 3^. 6d,
History of the English Institutions. By Philip v.
Smith, M.A., Fellow of King's College, Cambridge, y, 6d,
History of Modern English La\^. By Sir Roland Knyvet
Wilson, Bart., M.A., laie Fellow of Idn^s College, Cambridge,
y. 6d,
History of French Literature. Adapted from the French of
M. Demogeot, by C. Bridge. 3J. (kI,
(Recommended by the Intermediate Education Board for Ireland.)
History of the Romans to the Establishment of
Imperialism. By J. S. Reid, LL.M., Fellow and Assistant-
Tutor of Gonville arid Caius College, Cambridge; Classical Examiner
in the University of London.
[In preparation.
This work is intended to be used by the hig^her Forms in Public Schoob, and by Junior
Students in the Universities. It aims at exhibiting in outline the growth of the Roman
national life in all departments. Military history will not be neglected, but attention will
be particularly directed towards the political and social changes^ and the development of
law, literature, religion, art, science, and social life. Care will be taken to bring the
whole narrative into accord with the present state of knowledge, and also to present iStat
facts of Roman History in a form likely to interest the Students for whom the work is
intended.
WATERLOO PLACE, LONDON.
4
lo MESSRS, RIVINGTON'S [HI8T0BY.
Historical Biographies. Edited by the Rey. m.
Creighton, M. a., kUe Fellow and Tutor of Merlon College^ Oxford,
With Maps and Plans. Small 9fvo,
Simon de MontforL By M. Creighton, M.A. 2s. 6d,
The Black Prince. By Louise Creighton, 2j. 6il,
Sir Walter Ralegh. /?y Louise Creighton. With Portrait, y.
Oliver Cromwell. By F. W. Cornish, M.A. y. ed.
The Duke of Marlborough. By Louise Creighton. With
Portrait. 3j. 6d,
The Duke of Wellington. By Rosamond Waite. With
Portrait, y, 6d.
Crown 9^0. 6s,
A Handbook in Outline of the Political History of
England to 1881, chronologically arrange By Arthur
H. D. AcLAND, M.A., Christ Church, Oxford, and Cyril Ran-
SOME, M.A, Professor of Modem Literature and History, Yorkshire
College, Leeds,
Small ifz/o, IS. 6d.
A Skeleton Outline of the History of England,
being an abridgment of a Handbook in Outline of the Political
History of England. By Arthur H. Dyke Acland, M.A., and
Cyril Ransome, M.A.
Second Edition, Revised, Crown l&vo, p, 6d,
History of the Church under the Roman Empire,
A.D. 30-476. By the Rev. A. D. Crake, B.A,, Rector of Jfauen
Street, Ryde,
New Edition, \%nw, is, 6d,
A History of England for Children, ^^^georgedaws,
D.D., formerly Bishop of Peterborough,
WATERLOO PLACE, LONDON,
SraBNOB.] EDUCA TIONAL LIST. 1 1
SCIENCE
Crown Svo, 2s. 6d.
Elementary Course of Practical Physics. By a. m.
WoRTHiNGTON, M.A., F.R.A.S. Assistant-Master at Clifton
College,
Third and Enlarged Edition, With Illustrations, 2\5,
Physical Geology for Students and General
Readers. By a, II. Green, M.A., F.G.S., Professor of Geology
in the Yorkshire College of Science^ Leeds,
New Edition^ Revised, With Illustrations, Crown Svo, 2s. 6d,
An Easy Introduction to Chemistry. Edited by the
Rev. Arthur Rigg, M.A., and Walter T. Goolden, M.A.,
Lecturer in Natural Science at Tondridge School.
Second Edition. With Illustrations, Crown Svo, 5 j.
A Year's Botany. Adapted to Home and School Use. By
Frances Anne Kitchener. Illustrated by the Author.
Medium Svo,
Notes on Building Construction.
Arranged to meet the requirements of the syllabus of the Science
and Art Department of the Committee of Council on Education,
South Kensington Museum.
Part I.— FIRST STAGE, or ELEMENTARY COURSE. WUh
325 woodcuts^ los. 6d.
Part II. — COMMENCEMENT OF SECOND STAGE, or
ADVANCED COURSE. With 277 woodcuts, los. 6d.
Part III.— ADVANCED COURSE. With 188 woodcuts, 21s.
Report on the Examination in Building Construction, held by the Science
AND Art Department, South Kensington, in May, 1875. — " The want of a text-
book in this subj'ectt arranged in accordance tuith the published syllabus^ and therefore
limiting the students and teachers to the prescribed course^ has lately^ been well tnet by
a work published by Messrs. Jiivingtons, entitled ' Notes on Building Constniction,'
arranged to meet the requirements of the Syllabus of the Science and Art Department
of the Committee of Council on EducatioKf South Kensington. {Signed)
H. C. Seddon, AlATORi R.E.
yiine x8, Z875. llnstructor in Construction and JSsttmating at tk$
School of Military Engineering, Chatham,}
WATERLOO PLACE, LONDON,
12 MESSRS. RIVINGTON'S [MATHBKATXOB.
MATHEMATICS
Riuington's Mathematical Series
Small $V0. 3J. Without Answers^ 2s. 6d.
Elementary Algebra. By j. Hamblin Smith, M.A., of
GoniHlle and Caius College^ and late Lecturer in Classics at St, Peters
College^ Cambridge.
A Key to Elementary Algebra. Crovm 8zv. 9^.
Small %vo, 2s. 6d,
Exercises on Algebra. By], Hamblin Smith, m.a.
(Copies may be had without the Answers.)
Crown %vo, 8j. 6d.
Algebra, part II. By E. J. gross, M.A., Fellow of GonvUle
and Caius College, Cambridge, and Secretary to the Oxford and
Cambridge Schools Examination Board.
Small Svo. y, 6d.
A Treatise on Arithmetic By j. hamblin Smith, M.A.
(Copies may be had without the Answers.)
A Key to Arithmetic. Crown ^0. gs.
Small Svo. 4^. 6d.
Elementary Trigonometry. By j. Hamblin Smith, M.a.
A Key to Elementary Trigonometry. Crown Sz/o. ys. 6d.
Crown Svo. ^s. 6d.
Kinematics and Kinetics. By E. j. Gross, m.a.
Crown Svo. 4J. 6d.
Greometrical Conic Sections. By G. Richardson, m.a.,
Assistant-Master at Winchester College, and late Fellow of St, John^s
College, Cambridge.
Small Svo. y.
Elementary Statics. By J. Hamblin Smith, M.A.
Small Svo. 3 j.
Elementary Hydrostatics. By j. Hamblin Smith, m.a.
Crown Svo. 6s.
A Key to Elementary Statics and Hydrostatics.
By J, Hamblin Smith, M.A.
WATERLOO PLACE, LONDON,
MATHEMATICS.] EDUCATIONAL LIST. 13
Small %(vo, 3x. dd.
Elements of Geometry. By j. Hamblin Smith, m.a.
Containing Books I to 6, and portions of Books 1 1 and 12, of
Euclid, with Exercises and Notes, arranged with the Abbreviations
admitted in the Cambridge University and Local Examinations.
Books I and 2, limp cloth, \s, 6^., may be had separately.
Prescribed by the Council of Public Instruction for the use of the schools of Nova
Scotia; authorized for use in the schools o/ Manitoba; recommended by the University
of Halifax/ NoTia Scotia^ by the Council of Public Instruction^ Quebec; and authorized
by the Education Department^ Ontario.
Crcwn Svo. &f . 6d,
A Key to Elements of Geometry. By j. Hamblin
Smith, M.A.
Small ^0, is.
Book of Enunciations for Hamblin Smith's Geo-
metry, Algebra, Trigonometry, Statics, and Hydro-
statics.
Small Svo, ^s.
An Introduction to the Study of Heat By ], Hambun
Smith, M.A.
Contents.— General Effects of Heat— Thermometry— Expansion of Gases— Expansion
of Solids — Expansion of Liquids— Calorimetry — Latent Heat — Measure of Heat —
Diffusion of Heat : Radiation — Convection — Conduction — Formation of Vapour, Dew,
&c. ; Trade Winds, Ebullition, Papin's Digester, Spheroidal Condition — Congelation —
Measurement of Work — Mechanical Equivalent of Heat — Miscellaneous Exercises —
Appendix — Index.
Crown Svo. 6s.
The Principles of Dynamics. An Elementary Text-book
for Science Students. By R. Wormell, D.Sc, M.A., Head-
AlasUr of the City of London Middle- Class School,
Small Svo. 3^. (>d.
Army and Civil Service Examination Papers
in Arithmetic, including Mensuration and Logarithms, set in recent
Examinations for the Army, Woolwich, Cooper's Hill, Home Civil
Service, &c. With Arithmetical Rules, Tables, Formula; and
Answers, for the use of Students preparing for Examination. By
A. Dawson Clarke, B.A., St. John's College^ Cambridge.
New Edition^ Revised. Craivn Svo. 6j. 6^.
Arithmetic, Theoretical and Practical. By w. h.
GiRDLESTONE, M.A., ^ Christ's College^ Cambridge,
Also a School Edition. Small Svo, y. 6d,
WATERLOO PLACE, LONDON.
H
MESS/^S, RIVINGTON'S
[liATnr.
LATIN
Neu) EdUion, reuiseii, Crmon ^o, 3f . 6d.
First Latin Writer, comprising Accidence, the Easier Rules of
Syntax illustrated by copious Examples, and progressive Exercises
in Elementary Latin Prose, with Vocabularies. By G. L.
Bennett, M. A., Head-Master of the High School, Plymouth,
A Key for the use of Tutors only. Crown dfvo. $s. •
Contents. — Prefacb— Accidence— Exercises on the Syntax (270) : The Simple
Sentence ; The Compound Sentence : Adjectival Clauses, Adverbial Clauses, Substantival
Clauses — Latin-English Vocabulary — English-Latin Vocabulary.
Crown Svo, 2s, 6d,
First Latin Exercises. Being the Exercises, with Syntax
Rules and Vocabularies, from a "First Latin Writer." By G. L.
Bennett, M.A.
Crown ^0. IS. 6d.
Latin Accidence. From a "First Latin Writer." By G. L.
Bennett, M.A.
** The hook is a perfect model of what a
Latin Writer should be ^ and is sogradu-
eUed that front the beginning of a boy's
classical course it will serve mm through-
out till the end as a text-book for Latin
prose composition. The exercises^ too, are
so interesting in themselves, and take up
the different idiomatic peculiarities in
such an easy and natural way that the
piUnl almost insensibly^ comes to be master
of them, without hmnng them glaringly
thrust upon him in little detached sen-
tences, which, when mixed up in a nar-
rative, he fails, of course, to recognise.
We cannot speak too strongdy of this little
work, and we say to every classical teacher ,
if you introduce this work into your junior
class, you will require no other work
throughout till you come to the fifth or
sixth form, and pet haps not even then.
The book has our unqualified approbation.
We ought to mention, for the sake qf those
who may think of using the worn!, that
there are two sets of vocabularies, which
obviate the necessity of having recourse to
any Latin Dictionary. " — Schoolmaster.
Second Edition. Crown Svo. y. 6d.
Second Latin Writer. ByO. L. Bennett, M.A., Head-Master
of the High School, Plymouth,
This work, in continuation of the First Latin Writer, gives hints on writing Latin Prose
for Boys about to commence the rendering of continuous passages from English Authors
into Latin. There is a large Collection of Exercises, graduated according to their diffi-
culty, with Notes.
A Key for the use of Tutors only. Crown Svo. Ss,
*' This is one of the best introductions are well adapted to be taken up by the stu-
to Latin prose composition we hai>e seen.
The introductory remarks, the chapter on
tlte analysis of the Latin sentetice, the
observations on style, the, table of miscel-
laneous idioms, and the collection of exer-
cises for practice, furtiished with notes to
assist the student in points 7vhtch present
difficulties, are ail excellent. The pas-
sages used for translation into Lectin are
specimens of continuous narrative, and
dent who hasj'ustgone through the ordinary
Latin exercise books." — Schoolmaster.
''Mr. Bennett's Second Latin Writer
7vill be, or sJiould be, of very great ser-
Tice to stiuients 7vho have ticquired a fair
tnastery over the rudiments of the lang-
uage. The student who honestly ^vor^s
through this book will have tu^ired a
very great degree oj facility tn Latin
prose. '—Scotsman.
WATERLOO PLACE, LONDON.
LATIN.] EDUCATIONAL LIST, 15
New EdiHoHj revised. Crown 2ivo, 2s. 6d,
Easy Latin Stories for Beginners. By a u bennett,
M.A., Head-Master of the High School, Plymouth, With Notes
and Vocabularies. Forming a First Latin Reading Book for Junior
Forms in Schools.
A Key for the use of Tutors only. Crown %vo, 51.
Small Svo. 2s,
Selections from Caesar. The Gallic War. Edited,
with Preface, Life of Caesar, Text, Notes, Geographical and Bio-
graphical Index, and Map of Gaul, ^ G. L. Bennett, M.A.,
Head' Master of the High School, Plymouth.
Crown Svo, is, 6d,
First Steps in Latin. By F. Ritchie, M.A., Assistant-Master
at the High School, Plymouth,
'* Thanks for Ritchie's * First Steps in Latin* In my jtu^nent it is much sounder
and better than anything else you have published for young children learning Latin"
J. G. Cromwell,
St. Mark's College^ Chelsea.
Small Svo, is. 6d,
GradatlW, An Easy Translation Book for Beginners. By H. R.
Heatley, M.A., andH. N. Kingdon, B.A., Assistant-Masters at
Hillbrffiv School, Rugby.
The aim of this book is to provide translation for boys immediately on beginning Latin.
With this view care is taken that the beginner encounters no difficulty of Grammar or
Syntax without due warning.
Crown Svo, is. 6d.
The Beginner's Latin Exercise Book. Affording Practice,
oral and written, on I^tin Accidence. By C, J, S her will Da we,
B. A., Lecturer and Assistant Chaplain at St. Afark's College^ Chelsea.
m
New Edition. Crown Sfvo, 2s, 6d,
Latin Prose Exercises. For Beginners, and junior Forms of
wSchools. By R. Prowde Smith, B.A., Assistant- Master at
Cheltenham College,
Crown Svo. On a card, gd.
Elementary Rules of Latin Pronunciation. By
Arthur Holmes, M.A., late Senior Fellow and Dean of Clare
College^ Cambridge,
WATERLOO PLACE, LONDON.
. #:
i6 MESSRS. RIVINGTON'S [I^ATIK.
\%mo,
Latin 1 6Xt8, For use in schools, &c In stiiched wrapper^
THE AENEID OF VERGIL. Books I. II. III. IV. V. VII.
VIII. IX. 2d, each. Books VI. X. XI. XII. y. each.
THE GEORGICS OF VERGIL. Books L-IV. 2d. each.
THE BUCOLICS OF VERGIL. 2d,
VBtgil. The Bucolics, Georgics, and iEneid in One Volume. ClotA
2J. 6d.
CAESAR DE BELLO GALLICO. Books L V. VII. VIIL y. each.
Books IL III. IV. VI. 2d. each.
Caesar De BellO OalliCO. in One Volume. Cloth, IS, 6d,
Fifth Edition^ Revised, Crown ^0, 3J. 6^.
Progresme Exercises in Latin Elegiac Verse.
By C. G. Gepp, M.A., late Head-Master of King Edward VL
School, Stratford-upon-Avon.
A Key for the use of Tutors only. %vo, 5j.
Twelfth Edition. l2mo. 2s.
A First Verse Book, Being an Easy Introduction to the Mechan-
ism of the Latin Hexameter and Pentameter. By Thomas Ker-
CHEVER Arnold, M. A.
A Key for the use of Tutors only. i2tno, is.
Crown Svo. y. 6d.
An Elementary Latin Grammar Byj, hamblin smith,
M. A., ofGonviUe and Caius College^ and late Lecturer in Classics a*
St. Peter's College, Cambridge,
Twenty-fifth Edition, \2mo, 3^.
Henry's First Latin Book. By thomas kerchever Arnold,
A Key for the use of Tutors only. i2fno. is.
A New and Revised Edition, i2tno. 3J.
Arnold's Henry's First Latin Book. Byc, g. gepp, m.a.,
late Head-Master of King Edward VI. School, Stratford-upon-
Avon; Author of ^^ Progressive Exercises in Latin Elegies Verse.^'
A Key for the use of Tutors only. Crown ^0. 5.F.
M
%
WATERLOO PLACE, LONDON.
LATIN.] EUUCATIONAL LIST. 17
Twentieth Edition^ %vo, 6s, 6d,
A Practical Introduction to Latin Prose Com-
position. By Thomas Ke&chever Arnold, M.A.
A Key for the use of Tutors only, izmo, is, 6d,
A New and Revised Edition, Crown ^0, $s.
Arnold's Practical Introduction to Latin Prose
Composition. By G. Granville Bradley, D.D., Dean of
Westminster^ late Master of University College^ Oxford^ and formerly
Master of Marlborough College.
A Key for the use of Tutors only. 5^.
*' An Introduction has been prefixed containing three parts, two of which are new, th«
other much modified. The first of these is an explanation of the traditional terms by
which we designate the different ' parts of speech' in English or Latin. The exposition
is confined to the most simple and elementary points. This is followed by a few pages on
the Analysis of the Simple and Compound Sentence. Such logical analysis of the lan-
guage is by this time generally accepted as the only basis of intelligent grammatical
teaching, whether of our own or of any other language. I have followed Mr. Arnold's
example in prefixing some remarks, retaining so far as possible his own language, on the
Order of Words ; I have added some also on the arrangement of clauses in the Latin
Sentence. The matter for translation as comprised in the^ various exercises has been
almost entirely rewritten. I have not, after full consideration, taken what would have
been the easier course, and substituted single continuous passages for a number of sepa-
rate and unconnected sentences. 1 found that for the special purpose of the present
work, dealing as it does with such manifold and various forms of expression, the employ*
ment of these latter was indispensable, and I have by long experience convinced myself
of their value in teaching or studying the various turns and forms of a language which
differs in such innumerable points from our own as classical Latin."
Kxtraci from the Preface.
Crown %vo.
The Aeneid of Vergil. Edited, wUh Notes at the end, by Francis
Storr, M.A., C hief 'Master of lilodem Subjects at Merchant Taylors^
School.
Books I. and II. 2s. 6d, Books XI. and XII. 2s, 6d.
Small Svo, is. 6d,
Virgil, GeOrgiCS. book IV. Edited, with Ute, Notes, Voca^
btUary, and Index, by C. G. Gepp, M. A., late ffead-Mastir of King
Edwctrd VI. School, Stratford-upon-Avon, and Editor of ** Arnolds
Henrys First Latin Book," revised edition.
Small %vo. \s. 6d.
Selections from the Aeneid of Vergil, with Notes.
By G. ll Bennett, M.A., Head- Master of the High School^
Plymouth.
WATERLOO PLACE, LONDON,
1 8 MESSRS. RIVINGTON'S [X.ATXV,
Neu> Ediium^ Revised, Crown %fvo. 31. td.
Stories from Quid in Elegiac Verse, with Notes for
School Use and Marginal References to the Public School Latin
Primer. By R. W. Taylor, U.h.,Jlead- Master oJ Kelly College,
Tavistock, and late Fellow of St. yohn*s College, Cambridge.
New Edition^ Revised. Crown $zw. 2s. 6d.
Stories from Ovid in Hexameter Verse. K/feta-
morphOSeS. with Notes and Marginal References to the Public
School Latin Primkr. By R. W . Taylor, M.A.
JVew Edition, Revised, \2mo, 2s, 6d,
EclogCB OuidianCB. From the El<^ac Poems. With English
Notes. By Thomas Kerchever Arnold, M.A.
Small 9rvo, 2s.
Cicero de Amicitia. Edited, wUk Notes and an Introduction, by
Arthur Sidgwick, M.A., Tutor of Corpus Christi College,
Oxford; late Assistant- Master at Rugby School, and Fellow of Trinity
College, Cambridge,
C0NTRNT8. — Introduction: Time and Circumstances — Dedication — Scheme of the
Dialogue — Characters of the Dialoaues : 'I1ie Scipionic Circle— Pedigree of the ScipioA—
Conspectus of the Dialogue — Analysis. Laelius De Amiciti& — Notes— Scheme of the
$9Uhjunctive— Notes on the Readings — Indices.
Small %fvo. 3J. 6^.
C(2sar. De Bello Gallico. books liil Edued, -with
Preface, Introductions, Maps, Plans, Grammatical, Historical^ and
Geographical Notes, Indices, Grammatical Appendices, 6r*c., by J, II.
Merryweather, M.A., and C. C. Tancock, M.A., Assistant-
Mcuters at Charterhouse,
Book I. separately. 2s,
Crown Sz/o. 3^. 6d,
Exercises on ttie Elementary Principles of Latin
Prose Composition, with Examination Papers on the Ele-
mentary Facts of I^tin Accidence and S3mtax. By J, Ham bun
Smith, M.A., 0/ Gonville and Caius College, and late Lecturer in
Classics at St. Peter's College, Cambridge,
A Key. Crown ^0, $s.
Small &2Hf, is. 4d.
Easy Exercises in Latin Prose. By charles bigg, d.d.,
formerly Principal of Brighton College.
WATERLOO PLACE, LONDON.
.ULTIH.] EDUCATIONAL LIST, 19
A Latin-^English Diotionary for Jinihr J^0rm${tf
Schools. By C. G. Gepp, M. A. [In thi press.
This work aims at sappljring in a concise fonn and at a low cost all the information
required by boys in Middltt Class Schools, or in the Jnnior Forms of Pablic Schools.
Archaisms (with the exception of such as occur in the most commonly read authors),
words peculiar to Plautus, and words found only in Late or Ecclesiastical Latin, have
been excluded accordingly. On the other hand. Proper Names have been briefly yet
adequately treated in alphabetical order in the body of the work. . No effort has bwi
spared to ensure completeoess and acctuacyy ail references KiviBg heen V^Mfird fiDmlhle
latest and most approved editions of modem scholars. While everv legitimate aid has
been given to schoolboys, with whom the looking out a meaning is onen a very haphazard
process, it is hoped that the volume may be found a useful and handy companion to
many who seek to renew their acquaintance with the favourites of bygone days.
2fvo, On a card, IS, v. •.'• .v^\**^
Outlines of Latin Sentence Construction. By e. d.
Mansfield, M.A., Assistant* Masttr at CUfidn Collegt.
Second Edition, Crown %vo, *js, 6d, . ,
Classical Examination Papers. Edited, wuh Notes and
References, by P. J. F. Gantillon, M.A., Clctssical Master at
Cheltenham College. •/■ ^ >; ,9 ■ ! t^' > \ ^ 1
Or, interleaved with writing-paper, half-bound, lOf. 6d,
New Edition, re-arranged, with fresh Pieces and additional Eefercftces,
Cronvn Svo, 6s. 6d, \ ' ' '
Materials and Models for Latin Prose Composition.
Selected and arranged by J. Y. Sargent, 'M.,A^^Eel^w ofidTutcr
of Hertford College, Oxford, and T, F. Dallin, M.A., lafe Tutor
and Felloiv of Queen* s College, Oxford,
New Edition^ revised^ with additional pieces. Crown Svo, $s,
Latin Version (116) of Selected Pieces from Materials
and Models. By J. Y. Sargent, M.A.
May be had by Tutors only, on direct applicatiQi|^tQ^t|;*p.Plil?liabQr^^^*y
Small Svo. y. 6d.
Liuy. BOOK II. Edited, chiefly from the text of Madvig, with
Notes, Translations, and Appendices. By Henry Belcher, M. A.,
Master of the Matriculation Class, Ktng^s College School, London,
WATERLOO PLACE, LONDON. i
20 Af£SS/^S, RIVINGTON'S [liA^TIH.
SmaU 9w, 2s,
Selections from Books VIII. and IX. of Livy. with
Notes and Map. By E, Calvert, LX..D., St. John^s Collie,
Cambridge; and R. S award, M.A., Fdkw 0/ St, John^s CoUq;e,
Cambridge; Assistant-Master at Shrewsbury School,
m
Fifth EdUioH, i2mo. is.
Cornelius NepOS. with critical Questions and Answers, and an
Imitative Exercise on each Chapter. ByT.'K. Arnold, M.A.
Crown Svo,
Terenti ComOBdiCB. Edited byr.U Pamllon, M.A., Fet/aw
of New College^ Oxford,
ANDRIA ET EUNUCHUS. With Introduction on Prosody, ^.d^.
Or separately, ANDRIA. With Introduction on Prosody, y, 6d.
EUNUCHUS. y.
Crown %vo, 51.
JuUenaliS SatirW. thirteen satires. Edited by G.
A, SiMCOX, M.A., Fellow of Queen's College^ Oxford.
Crown ^0, y. 6d. ^
Persii Satires. Edited by a. Pretor, M. a., of THnUy ColUge,
Cambridge,
Crown Svo, 7j. 6d,
Horati Opera. By J. M. Marshall, M.A., Under-Master at
Dulwich College,
Vol. I.— the ODES; CARMEN SECULARE, and EPODES.
Also separately, THE ODES. Books I. to IV. is. 6d. each.
Crown 9t/o,
Tacit i Histories. Edited by W. H. Smcox, M.A., Fdlmv of
Queen's College^ Oxford,
Books I. and II., ds. Books III., IV., and V., ts.
WATERLOO PLACE, LONDON,
.] EDUCATIONAL LIST. 21
GREEK
Second Edition, Revised. Crtrwn 2ivo, 3x. 6d,
A Primer of Greek Grammar, with a Preface i^ john
Percival, M.A., LL.D., President of Trinity College, Oxford;
late Head-Master of Clifton ColUige,
This book is in use at Eton, Rugby, Clifton, Edinbuigh, Rossall, Uppingham,
Felstead, &c.
Or separately, crown $vo. 2s. 6d,
Aooidence. By Evelyn Abbotf, M.A., LL.D., Fdlow and Tutor
gf BaUiol ColUge, Oxford; and £. D. Mansfield, M.A.,
Assistant' Master at Clifton College,
Crown $vo. is, 6d,
Syntax. By £. D. Mansfisld, M.A., Assistant-Master at Clifton
College,
This outline of the chief Rules of Greek Syntax, which is intended as a K^uel to the
" Primer of Greek Accidence," lays no claim to originality of treatment. Tne Editor
has freely consulted the usual authorities, especially tne well-known " Greek Moods and
Tenses," and the later *' Elementary Greek Grammar," of Professor W. W. Goodwin,
and has only aimed at stating Rules simply and concisely, and so grouping them as to
indicate general principles and prepare the beginner for the use of a fuller treatise. He
is largely indebted in tne first part di the Syntax to material kindly placed at his dis-
posal by Mr. Evelyn Abbott^ which, however, has for teaching purposes been thrown
uto a shape for which the Editor alone is responsible.— ^jr/nK/ fivm tht Preface,
Contents. — Part. I.— Agreement. The Cases. Accusative. Genitive. Dative.
Prepositions. Article. Pronouns. Tenses. Notes on the Tenses. Moods. Infinitive.
Participle. Verbal Adjective. Negatives od and A^i^. Conjunctions and Particles.
Conjunctions. Participles. Part II. — The Simple Sentence. Direct Statement.
Direct Command. Expression of a Wish. Direct Question. The Compound Sentence.
Substantival Clauses : Indirect Statement — Indirect Command — Indirect Question. Ad-
jective Clauses. Adverbial Clauses: Final — Consecutive — ^Temporal — Conditional —
Concessive — Casual. Adjectival Clauses with Adverbial force. Further Roles for
Indirect Speech. Dependent Clauses in Indirect Speech.
Part I. Crown Svo, y. 6d,
A Practical Greek Method for Beginners, Being a
Gnuluated application of Grammar to Translation and Composition.
By F. Ritchie, M. A., and K H. Moore, M. A., Assistant-Masters
at the High School, Plymouth.
Containing the Substantives, Adjectives, Pronouns, and Regular Piure Verbs, with
exercises (English-Greek and Gredc-English), introducing the main rules of Syntax of
the Simile Sentence.
The aim of this book, which is at once a Grammar and Exercise Book, is to secure an
uniform method of teaching Grammar, and to afford abundant practice in inflexion, &C|
at the time that Uie Grammar is being learnt.
PART II. in preparation.
IVATEKLOO PLACE, LONDON.
22 MJSSS/iS. R/y/NGTON*S i^mmoLkj
Second Edition, Reyised, Crqwn 9fVo. 31. 6d,
A First Greek Writer: sy Arthur sidgwick, m.a., tioot
of Corpus ChrisH CoiUge, Oxford, laU Assutan^Master at Rugbv
School, and Fellow of Trinity CoUege, Cambrulge.
A Kby for the use of Tutors only. Crown 2a/o, 5/.
General Contents. Hints on Writing Greek. The Articles. Pronouns. Attrac-
tion. Adjectives. Cases. Infinitive. Participle. Tense Idioms. Adverbs. Dramatic
Particles. About 120 Exercises, with special and general vocabularies.
TTiird Edition, Revised, Crown $vo, 5 j.
An Introduction to Greek Prose Composition, with
Exercises. By Arthur Sidgwick., M.A.
A Key for the use of Tutors only. 5^.
Sixth Edition, l2mo, ^s.
The First Greek Book. On the pUm of Hentys First Latm
Book. By Thomas Kerchever Arnold, M.A.
A Key for the use of Tutors only. \2mo, is, 6d,
New Edition, Revised, Crown ^o,
Arnold's First Greek Book. By francis david morice,
M.A., Assistant'Mctster at Rugby School, and Fellow of Qmetn^s
College, Oxford, [In preparatt^H,
Third Edition, Imperial l6mo, &r. 6d,
Maduig's Syntax of the Greek Language, espe-
QiallU of the AttiO Dialect For the use of Schools.
Ediud by Thomas Kerchever Arnold, M.A.
Recommended by the Cambridg;e Board of Classical Studies for the
Classical Tripos.
Ninth Edition^ $vo,^ $s. 6d,
A Practical Introduction to Greek Accidence.
By Thomas Kerchever Arnold, M.A.
Thirteenth Edition. 8w. $1. 6d,
A Practical Introduction to Greek Prose Com^
position. By Thomas Kerchever Arnold, M.A.
A Key for the use of Tutors only. i2mo, is. 6d.
IVATERLOO PLACE, LONDON,
GObBBK.] EDUCATIONAL UsT: n
New Edition, Revised. Crown %uo» y, dd,
Arnold's Practical Introduction to Greek Prose
Composition, By Evelyn Abbott, M.A., LL.D., Fellow and
Tutor of Balliol College j Oxford.
A Key for the use of Tutors only. Crown Svo, y, 6d,
'* I have endeavoured to keep everything that seemed of value, and often I have
adhered to the words of the explanations because I wished to preseiftre, as far so possible,
the continuity of the IxJok. Hut I have added illustrations, altered the order of the
sections, and indeed rearranged the matter of the sections themselves. Mrherever I thought
that, by doing so, I could gain in clearness or simplicity ; I have also rewritten, tUuosi'
entirely, the sentences in the exercises. The lists of accents, irregular verbs, &c., which
Mr. Arnold prefixed to his book, I. have omitted, because all that is required on these
matters can be obtained from the Greek Accidence ; and I have also omitted the refer-
ences to grammars now no longer in general use among scholars, the list of particles, and
the (questions on syntax at the end of the exercises. The table of idioms is retained, with
alterations, and references to it are given in the exercises — though I would strongly re-
commend the student to learn this table by heart, and so render reference unnecessary.
The vocabulary is, I believe, nearly complete, and the index of matters will serve as an
independent table of references, whenever those given in the text are insufficient."
Extract from the Pre fact.
Crown ^vo. 4s, 6d. ■■.<'.'
Elements of Greek Accidence. By evelyn abbott,
M.A., LL.D., Fellow and Tutor of Balliol College^ Oxford.
Crown Svo. 4s. 6d.
An Elementary Greek Grammar By j. hambun
Smith, M.A., of GonvUle and Coins College, and late Lecturer in
Classics at St. Peter's College, Cambridge.
Cloth limp, Svo. is.
A Table of Irregular Greek Verbs, classified
according to the arrangement of Curtius's Greeif
Grammar. By Francis Storr, M.A., Chief Master of Modern
Subjects at Merchant Taylors^ School, and late Assistant' Master at
Marlborough College.
Cloth limp, Svo. 6d.
Elementary Card on Greek Prepositions. By Rev.
E. Priestland, M.A., Spondon House School, Derbyshire.
Crown Svo. gd.
A Short Greek Syntax. Extracted from "xenophon's
Anabasis, with Notes." By R. W. Taylor, M.A., Head^MastttL
of Kelly College, Tavistock.
IVATERLOO PLACE, LONDON. i
U Af£SSRS. RIVINGTON'S [GBEfilL
Second EdiHam Revised. Small %vo. is, 6d,
Zeugma ; or, Greek Steps from Primer to Author.
By the Rev. Lancelot Sanderson, M.A., Principal of Elstree
School^ late Scholar of Clare Coll^e^ Cambrui^ ; and the Rev. F. B.
FirmAn, M.A., AsHstdnt'Mdster at S&nroyd Schodl, Cobhafn, Uue
Scholar of Jesus College^ Cambridge,
Second Ediiioh, Cr&wn ^o. Js, 6d
Ctassical Examination Papers. Edited, wuh Ndtes and
R^ereiuesy by P. J. F. Gantillon, M.A., Classical McLster at
Chdtenham College.
Or interleaved with writing-paper, half-bound, lOf. 6^.
Second Edition, containing Fresh Pieces and additional References.
Crown ^vo, 5J.
Materials and Models for Greek Prose Com-
position. Selected and arranged by J. Y. Sargent, M. A., Fellow
and JlOorof Hertford College, Oxford; andT. F. Dallin, M.A.,
Tutor, late Fellow, of Queen* s College, Oxford,
Crown Svo, Js, 6d,
Greek Version of Selected Pieces from Materials and
Models. By J. Y. Sargent, M.A.
May be had by Tutors only, on direct application to the Publishers.
Crown ^0. 2s,
lophon : An Introduction to the Art of Writing
Greek Iambic Verses. By the writer of ^'Mnces" and
'' LucretUis."
Fifth Edition, Crown Svo, is, 6d,
Stories from Herodotus. The Tales of Rhampslmtus and Poly,
crates, and the Battle of Marathon and the Alcmaeonidae. In Attic
Greek, Edited by J, Surtees Phillpotts, M. A., Head-Master of
Bedford Grammar School,
New Edition, Small Svo, y. 6d,
Selections from Lucian. with English Notes. ^>/Evelyn
Abbott, M.A., LL.D., Fellow and Tutor of Balliol Coll^e^ Oxford.
Ml II, ' ■— --■ - ^ -. —
WATERLOO PLACE, LONDON,
.] EDUCATIONAL LIST. 25
Small %vo, IS. 6d. each.
Scenes from Greek Plays, rugby edition.
Abridged and adapted for the use ef Schools^ by Arthur Sidgwick,
M.A., TkiUor ef Corpus Christi CoHeffe^ Oxfij/rd^ late Assistant'
Master at Rugby School^ and Fellow of Trinity College^ Cambridge.
Aristophanes.
THE CLOUDS. THE FROGS. THE KNIGHTS. PLUTUS.
Euripides.
IPHIGENIA IN TAURIS. THE CYCLOPS. ION
ELECTRA. ALCESTIS. BACCHiE. HECUBA
Third Edition, Crorum Svo, y, 6d.
Stories in Attic Greeli. Fonnmg a Greek Reading Book for
the use of Junior Forms in Schools. With Notes and Vocabulary.
By Rev. Francis David Morice, M.A., Assistant-Master at
Rugby School f and Fellow of Queen's Coll^^ Oxford.
Crown Sivo,
Ttie Anabasis of XenOptlOn. Edited, with Preface, intro-
duction, Historical Sketch, Itinerary, Syntcuc Rules, Notes, Indices,
Vocabularies, and Maps, by R. W. Taylor, M.A., Head-Master of
Kelly College, Tavistock, and late Fellow of St. John's CoUege, OflW»-
bridge.
Books L and II. y. (hI. Books IIL and IV. 3J. dd.
Also separately, Book I., 2s. 6d. ; Book II., 2s.
Crown Svo, 2s. 6d.
XenOptlOn 'S AgeSilaUS. Edited, with syntax RuUs, and Refer-
ences, Notes and Indices, by R. W. TAYLOR, M.A.
Small ^0. 2s.
Xenoption 's Memorabilia, book l, with a few omissions.
Edited, with an Introduction and Notes, dytheRtv. C. E. MoBERLY,
M. A., formerly Scholar of Balliol CoUege, Oxford.
Small ^0. 2s.
Alexander the Great in the Punjaub. Adapted from
Arrian, Book V. An Easy Greek Reading Book. Edited, with
Notes and a Map, by the Rev. C. E. Moberly, '^.K., formerly
Scholar of Balliol College, Oxford.
WATERLOO PLACE, LONDON
B*
26 Jii£SSJ^S. RIVINGTON'S [OILSBK.
SntiUl 8»<7. ^ ,
Homer's Iliad. E^tOd^wUh NoUs at the end for thMUu of junior
Students, by Arthur Si dg wick, M.A., Tutor of Corpus Christi
College, Oxford; late Assistant^ Master at Rugby School, and Fellow
of Trinity College, Cambridge,
Books I. and II. 2s. 6d.
Contents. — Preface — Introduction — The Language of Homer — The Dialect — Forms
— Syntax — General Text, Books I. and II. — Notes — Indices.
Book XXI. is, 6d. Book XXII. is, 6d,
Small Szfo, 2s.
Homer without a Lexicon, for Beginners, iliad.
Book VI. " Edited, with Notes giving the fneanings of all the less
common words, by^, SuRTSES Phillpotts, M. A., Hecut' Master of
Bedford Grammar School,
Fifth Edition, i2mo, y. 6d.
Homer for Beginners, iliad, books liii. with English
Notes. By Thomas Kerchever Arnold, M. A.
Fiy^h Edition, i2mo, I2s.
The I Had of Homer. with EngUsh Notes and Grammatical
References. By Thomas Kerchever Arnold, M.A.
Crown Svo, dr.
The Iliad of Homer, books I.-XII. From the Text of Din-
dorf. With Preface and Notes. By S. IL Reynolds, M.A., late
Fellow and Tutor of Brasenose College, Oxford,
New Edition, i2mo, gs.
A Complete Greek and English Lexicon for
the Poems of Homer and the Homeridce. ByG.
Ch. Crusius. Translated from the German, Edited by T, K.
Arnold, M.A.
Crown Svo, 4^. 6d.
Isocratis Oratiohes. ad demonici/m et panegyricus.
Edited by John Edwin Sandys, M.A., Fellow and Tutor of St.
John^s College, Cambridge, and Public Orator of the University,
WATERLOO PLACE, LONDON,
.] EDUCATIONAL LIST. 27
&/^. 1 6 J.
Hellenica. A Collection of Essays oh Greek Poetry,
Philosophy, History, and Religion. Ediud by Evelyn
Abbott, M.A., LL.D., Fellow and Tutor of Balliol College, Oxford.
Contents. — Aeschylus. E. Myers, m.a. — The Theology and Ethics of Sophocles.
E. Abbott, M.A., LL.D. — System of Education in Plato's Republics. R. L. Nettleship,
M.A. —Aristotle's Conception of the State. A. C. Bradley, m.a. — Epicurus, W. L.
Courtney, m.a.— The Speeches of Thucydides. R. C. Jebb, m. a.,lud.— Xenophon. H» Ow
Dakyns, m.a. — Polybius. J. L. S. Davidson, m.a.— Greek Oracles. F. W. H. Myers, m.a.
^VO, I&f.
Myths of the Odyssey in Art and Literature.
Illustrated with Outline Drawings. By]. E. Harrison.
^0. i&r.
The Antiquities of Greece, the state. Translated from
the German of G. F. Schoemann. By E. G. Hardy, M.A.,
Head-Master of the Grammar School , Grantham ; and J. S. Mann,
M. A., Fellow of Trinity College^ Oxford,
Crown %vo,
Herodoti HistOria. Edited by H. G. woods, m.a., Fellmv and
Tutor of Trinity College, Oxford.
Book I. dr. Book II. 55.
\2mo.
Demosthenes. Edited, with English Notes and GrammsHcai
References, by Thomas Kerchever Arnold, M.A.
OLYNTHIAC ORATIONS. Third EdUion. 3J.
ORATION ON THE CROWN. Second Edition. 4?. 6^.
Crown Sfvo. ^s.
Demosthenis Orationes Priuatce. de corona.
Edited by Arthur Holmes, M.A., late Senior Fellew and Dean
of Clare College, Cambridge.
Crown Sivo.
Demosthenis Orationes Pub/icce. Edited by g. h.
Heslop, M.A., late Fellow and Assistant-Tutor of Queen^s College^
Oxford; Head- Master of St. Bees.
OLYNTHIACS, 2s. 6d. ) - r^ ^r ^ ^ ..
PHILIPPICS, y. \ ^'■' '" ^"^^ Volume, ^. 6d.
DE FALSA LEGATIONE, 6s.
WATERLOO PLACE, LONDON.
i
a8 JIfESSJiS. RIVINGTON'S
Second EdUioftj Revised and Enlturmd. Croum 8zv. lOf. 6^.
An Introduction to Aristotle's Ethics, books l-iv.
(Book X., c. vL-ix. in an Appendix). With a Continaons Analjrsis
and Notes. Intended for the use of Beginners and Junior Students.
By tAeRey. Edward Moore, B.D., /yinctpcd o/St, Edmund Ha//,
and /ate Fe//cw and Tutor of QueetCs Cot/ege, Oxford,
SmcU/^vo. \s, 6^.
Aristotelis Ethioa Nicomactiea. Edidit, em«idavit,
crebrisque locis parallelis e libro ipso, aliisque ejusdem Auctoris
scriptis, illustravit Jacobus E. T. Rogers, M. A.
Interleaved with writing-paper, half-bound. 6s.
Second EdUion. Crown Svo, 3J. 6d,
Selections from Aristotle's Orgahon. Edited dy jom^
R. Magrath, D.D., Provost of Queen^s Co//egey Oxford,
i2mo.
Sophocles, Edited dy T. K. Arnold, M.A., Archdeacon Paul,
and Henry Brown, M.A.
AJAX. y, PHILOCTETES. y.
OEDIPUS TYRANNUS. 4s,
Crown Sz/o.
Sophoclis TragoBdicB. Eduedby r. c. jebb, m.a., ll.d..
Professor of Greek at the University of G/asgoiVy /ate Fe//ow and
Tutor of Trinity Co/Zege, Cambridge.
ELECTRA. 3J. ed, AJAX. y. 6d,
Crown Sz/o,
Aristophanis ComoBdice. Edited by w. c. green, m. a.,
/ate Fe//ow of King's Co//ege, Cambridge; Assistant - Master at
Rugby Schoo/,
THE ACHARNIANS and THE KNIGHTS. 4f.
THE CLOUDS, is, 6d, THE WASPS, 3J. 6d.
Crown 82/(7. 6s,
Thucydidis Historia, books i. and ii. Edited by charlks
Bigg, D.D., /ate Senior Student and Tutor of Christ Churchy
Oxford ; forfner/y PrincipcU of Brighton Co//ege,
Crown Svo, 6s,
Thucydidis Historia. books hi. and iv. Edited by g. a.
SiMCOX, M. a., Fe//ow of Queen's Co/Zege, Oxford,
WATERLOO PLACE, LONDON,
GtaEEX.] EDUCATIONAL LIST. 29
Fifth Edition, Svo, 21s,
A Copious Phmseologioal Eng/ish-Greek Lexicon.
Flmnded on a work prepared by J. WT Fradersdorff, Ph.D., late
Professor of Modem Languages^ Queen* s College^ Belfast, Revised,
Enlarged, and Improved by 'fHOViK'& Kerchever Arnold, M.A.,
and ilENRY BROWNE, M.A.
Third Edition. Crown Svo. 2s, 6d,
Short Notes on the Greek Text of the Gospel of
St, Mark, By J. Hamblin Smith, M. a., of Gonville and
Caius College, Cambridge,
Crown Sivo, 4^. 6d,
/Votes on the Greek Text of the Acts of the
Apostles. By J. Hamblin Smith, M.A., of Gonville and Cairn
College, Cambridge,
Crown Svo, 6s.
/Votes on the Gospel According to 8. Luke.
By the Rev. Arthur Carr, M. A., Assistant-Master at JVelUngtom
College, late Fellow of Oriel College, Oxford,
New Edition, 4 vols, $vo. 102s,
The Greek Testament WUh a Cntically Revised Text ; a
Digest of Various Readings ; Marginal References to Verbal and
Idiomatic Usage ; Prolegomena ; and a Critical and Exe^etical
Commentary. For the use of Theological Students and Ministers.
By Henry Alford, D.D., late Dean of Canterbury.
The Volumes are sold separately, as follows : —
Vol. I.— THE FOUR GOSPELS. 28^.
Vol. II.— ACTS TO 2 CORINTHIANS. %\s.
Vol. III.— GALATIANS to PHILEMON, i&f.
Vol. IV.— HEBREWS to REVELATION. 32^.
New Edition, 2 vols. Imperial ^0, 60s,
The Greek Testament with Notes, introductions, ana
Index. By Chr. Wordsworth, D.D., Bishop of Lincoln,
The Parts may be had separately, as follows : —
THE GOSPELS. 165.
THE ACTS. %5.
St. PAUL'S EPISTLES. 23J.
GENERAL EPISTLES, REVELATION, and INDEX. \6s.
WATERLOO PLACE, LONDON,
30 MESS/US. RIVINGTON'S [OBEBK.
CATENA CLASSICORUM
Crown $z/o.
Aristoplianls Comoedide. By w. c Grekn, m.a.
THE ACHARNIANS AND THE KNIGHTS. 4*.
THE WASPS, y. 6d. THE CLOUDS, y, &/.
Demostlienis Orationes Publicae. ByG.lL Heslop, m.a.
THE OLYNTHIACS. 2s. M. \ - ru. ^r ^ . ^v
THE PHILIPPICS, y. ] ®^' ^ ^^ Volume, 4s. 6d.
DE FALSA LEGATIONE. 6s.
Demosthenis Orationes Privatae. de corona. By
Arthur Holmes, M.A. 5^.
H^rodoti Historia. By H. G. Woods, M.A.
Book I., 6s. Book IL, $s.
Homeri Ilias. By S. H. Reynolds, ^LA.
Books I. -XII. 6s.
Horati Opera, By j. M. Marshall, M.A.
THE ODES, CARMEN SECULARE, and EPODES. 7s. 6<L
THE ODES. Books I. to IV. separately, u. 6d. each.
Isocratis Orationes, ad demonicum et panegyricus.
^^ John Edwin Sandys, M.A. 4f. 6d.
Juvenalis Satirae. By G. A. Simcox, m.a. 5^.
Persii Satirae. By A. Pretor, m.a. 3^. 6d.
Soplioclis Tragoediae. By R. c. Jebb, m.a.
THE ELECTRA. y. 6d. THE AJAX. 35. 6d.
Taciti Historiae. By W. H. Simcox, M.A.
Books I. and II., 6s. Books III. IV. and V., 6s.
Terenti Comoediae. andria and eunuchus. with
Introduction on Prosody. By T. L. Papillon, M.A. 41. 6^
Or separately.
ANDRIA. With Introduction on Prosody, y. 6d.
EUNUCHUS. 3^.
Thucydidis Historia. books I. and IL By Charles Bigg,
D.D. dr.
Books III. and IV. By G. A. Simcox, M.A. dr.
WATERLOO PLACE, LONDON.
MVINITT.] EDUCATIONAL LIST. 31
DIVINITY
Small 8t/<?. 3J. 6</. each. Or each Book in Fhe PaHs^ is, each Part,
Manuals of Religious Instruction, mm b,
John Pilkington Norris, D.D., Archdeacon of Bristol, Canon
Residentiary of Bristol Cathedral^ and Examining Chaplain to the
Bishop of Manchester.
The Old Testameht. [ The New Testament.
The Prayer Book.
Cheap Edition, Small Svo, is, 6d, each.
Keys to Christian Knowledge. ByiA^^.j.n.
Blunt, M. A., Editor of the ^*' Annotated Book of Common Prayer"
The Holy Bible. | The Church CatechJ^lBu
The Book of Common | Church History, Ancient.
Prayer. I ChurchHlstory, Modern.
jPj/John Pilkington Norris, D.D., Archdeacon of Bristol.
The Four Gospels. . | The Acts of the Apostles*
iSfHO. IS. 6d.
Easy Lessons Addressed to Candidates for Con-
firmation. -5?y John Pilkington Norris, D.D.
JSTetv Edition. Small ^o, is. 6d, •
A Manual of Confirmation, with a Pastond Letter instruct-
ing Catechumens now to prepare themselves for their First Com-
munion. By Edward Meyrick Goulburn, D.D., Dean of
Norwich,
idmOy IS. 6d. ; Paper Covers, \s. ; or in Three Parts , 6d. each.
The Young Churchman's Companion to the Prayer
Booh, By the Rev. J. W. Gedge, M.A., Diocesan Inspector of
Schools for the Archdeaconry of Surrey.
Part I.— MORNING and EVENING PRAYER, and LITANY.
Part II.— BAPTISMAL and CONFIRMATION SERVICES.
Part III.— THE HOLY COMMUNION.
Crown %vo, 'js. 6d,
Some Helps for School Life, sermons preached at Clifton
College, 1^2-1879. ^y J- Percival, M.A., LL.D., President
of Trinity College, Oxford, and late Head-Master of Clifton College,
fVATERLOO PLACE, LONDON.
32 MESSRS. RIVINGTON'S [DIVIKITT,
New Ediium. Small %iM>, y. 6d,
Household Theology, a Handbook of Religious information
respecting the Holy Bible, the Prayer Book, the Churchy the
Ministry, Divine Worship, the Creeds, &c &c. By tJWRev, Johh
Henry Blunt, M.A., F.S.A., Editor of The AnnataUd Book of
Common Prayer ^^^ dr»^. ^c.
Second EdiHon^ Revised, Crown Szw, 7 j« 6d,
Rudiments of Theology, a First Book for students. By
John Pilkington Norris, D.D., Archdeacon of Bristol, Canon
Residentiary of Bristol Cathedral ^ and Examining Chaplain to the
Bishop of Manchester,
Craifjn Siz/o, Js, 6d,
ECCleSia AngliOana. a History of the church of Christ in
England from the earliest to the present times. By Arthur
Charles Jennings, M.A., yesus College^ Cambridge; Vicar of
WhUtlesford,
Second Edition, \%mo, is, 6d,
The Way of Life. a Book of Prayers and Instruction for the
Young at School. With a Preparation for Holy Communion.
Combed by a Priest. Edited by the Rev. T. T. Carter, M.A.
Crown i6mo. Cloth limp, is. 6d.
A Manual of Devotion, chiefly for the Use of
Schoolboys, By William Baker, D.D., Head-Master of Mer-
chant Taylors' School. With Preface by J, R. WoODFORD, D.D.,
Lord Bishop of Ely,
Small Svo, is.
Church Principles on the Basis of the Church
Ctttechism. For the use of Teachers and the more Advanced
Classes in Sunday and other Schools. Z?y John Macbeth, LL.D.,
Rector of Killegney^ one of the Examiners under the Board of
Editions Education of the General Synod of the Church of Ireland.
Crown ^0. \s. Cloth limp^ is. 6d.
Study of the Church Catechism. Adapted for use as a
Class Book. By C. J. Sherwill Dawe, M.A., Lecturer and
Assistant'Chaplain at St. Marias College^ Chelsea,
WATERLOO PLACE, LONDON,
dl^ftltAlt.] kDUCAtlONAt LIST. 33
GERMAN
New Edition^ Revised, ijto, y, 6d,
A German Accidence for the Use of Schools. By
J. W. J. Vkcquer AY, AsHsUmt'Master at Rug^ SchikU.
Crown Siw, 2s,
First German Exercises. Adapted to Vecqueray's "German
Accidence for the Use of Schools." By £. F. Grxnfell, M.A.,
late AssistarU'Master at Rugby School,
CroTvn 8tv. 2s, 6d,
German Exercises. Part 11. wi* Hints for the Translation
of English Prepositions into German. Adapted to Vecqueray's
"German Accidence for the Use of Schools.'' By £. F. Grenfell,
M. A., Ic^ Assistant-Master at Rugby School,
Crown ^^0, 4^. 6d,
Selections from Hauff's Stories, a First German
Reading Book. Edited by W. E. Mullins, M. A., Assistant- Master
at Marlborough College^ and F. Storr, M. A., Chief-Master of Modem
Subjects in Merchant Tc^lori School.
AlsOy separately, crown $vo, 2s,
Kalif Stork and The Phantom Crew.
Eighth Edition, i2mo, ^s. 6d;
The First German Book. By t. k. Arnold, m.a., and
J. W, Fradersdorff, Ph.D. Key, i2fno, 2s, 6d,
Crown $z/o. 2s. 6d.
LeSSing'8 Fables. Arranged in order of difficulty. A First
German Reading Book. By F. Storr, M.A., Chief' Master of
Modem Subjects in Merchant Taylori^ School^ and UUe Aesisiamt*
Master in Marlborough College,
Crown Svo, ys. 6d.
Goethe 'S Faust, part I. Text, with English Notes, Essays, and
Verse Translations. By E. J. Turner, M.A., and E. D. A.
MORSHEAD, M.A., Assistant-Masters at Winchetter College,
Crown Sfvo.
WATERLOO PLACE, LONDOI^.
34 MMSSSS. RIVtNGTON'S [FBKNOtti
FRENCH
New Edition, Small $V0. is,
A Graduated French Reader, with an introduction on
the Pronunciation of Consonants and the Connection of Final
Letters, a Vocabulary, and Notes, and a Table of Irregular Verbs
with the Latin Infinitives* By Paul Barbier, on^ 9f the Modem
Language Masters at the Manchester Grammar School^ and
Examiner to the Intermodiate Education Board of Ireland, etc.
Crown Svo.
The Campaigns of Napoleon. The Text nn French) jh^i
M. Thiers^ ** J/istoire de la Rholution Franfaise,'^ and ^^ JOstoire
du Comulat et de rEmpire," Edited, with Endish Notes and
Maps, for the use of Schools, by Edward E. Bowen, M.A.,
Master of the Modern Side, Harrow School,
ARCOLA. 4J. 6d, MARENGO. 4r. (>d,
JENA, y, 6d, WATERLOO. 6j.
New Editions, Crown S^o, y, 6d, each.
Selections from Modern French Authors. Edited,
with English Notes and Introductory Notice, by Henri Van Laun,
Translator of Taine*s History of English Literature.
HONORS DE BALZAC. H. A. TAINE.
Small 9vo, 2J.
La Fontaine's Fables, books l and il Et^tcd, wuh
English Notes at the end, by Rev. P. Bowden- Smith, M.A.,
Assistant-Master at Rugby School,
Sixth Edition, i2mo, ^s, 6d,
The First French Book. By t. k. Arnold, m.a.
Key, izmo, zs, 6d,
SmaU 8ivo, 2s, 6d,
The Bengeo French Grammar, a lusage des Ecoies
Pr^paratoires. Far Emile de Tuetey, B.S.
WATERLOO FLACE, LONDON,
UISOBLLANBOVS.] EDUCATIONAL UST. 35
MISCELLANEOUS
WUh Maps, Small %vo.
A Geography, Physical, Political, and Descriptive.
For Beginners. By L. B. Lang. Edited by the Rev. M,
Creighton, M.A., late Fellow and Tutor of Merton College,
Oxford.
Vol. I. THE BRITISH EMPIRE. 2s, ed.
Part I. The British Isles, is. 6d. Part II. The British Possessions, is. 6d.
Vol. II. THE CONTINENT OF EUROPE. [In the press.
Vol. III. ASIA, AFRICA, AND AMERICA. [In pnparation.
Small ^0, 2s, 6d, each part.
Modern Geography for the Use of Schools.
By the Rev. C. E. M09ERLY, M.A., formerly Scholar of BaUiol
College, Oxford,
Part I. NORTHERN EUROPE.
Part II. THE MEDITERRANEAN & ITS PENINSULAS.
Crtnvn Svo, 3^. 6d,
At Home and Abroad ; or, First Lessons in
Geography. By j. k. laughton, m. a., f.r.a.s., f.r.g.s.,
Mathematical Instructor and Lecturer at the RoycU Naval College,
Second Edition, Crown Sfvo, 2s, 6d,
The Chorister's Guide. By w. a. barreti, mus. Bac.
Oxon., Vicar-Choral of St, Paulas Cathedral^ Author of ** Flowers
and Festivals " drv.
Crown Svo, 2s. 6d,
An Introduction to Form and Instrumentation.
For tlie use of Beginners in Composition. By W. A. Barr£T1\
Mus. ^ac. Oxon., Vicar- Choral of St. PauTs Cathedral,
Sixth Edition. . l2mo. 'js. 6d,
The First Hebreu) Book. By t. k. Arnold, m.a.
Key, I2m0y y. 6d.
WATERLOO, fLACE, LQNDQN. ^
36 MESSRS. RIVINGTOJfPS
BOOKS
SUITABLE FOR SCHOOL LIBRARIES OR PRIZES.
With numerotis Illustratums* Crown 2/vo. 5^.
Sacred Allegories. The Shadow of the Cross— The Distant
Hills— The Old Man's Honw— The King's Messengers* By the
Rev, William Adams, M.A., IcUe Fdlaw of Merton College^
Oxfrrd.
The Four Allegories may be had separately, with Illustrations^
161110. is, each.
Also a Presentation Edition, with numerous Illustrations. Crown
4to. I2s, 6d,
Crown %7M), 3^. 6^.
Edwy the Fair; or, The First Chronicle of .Sscen-
dune. A Tale of the Days of Saint Dunstan. By the Rev. A. D.
Crake, B.A.
Crown %vo. 3J. 6d,
Alfgar the Dane ; or, The Second Chronicle of ^s-
ceadune. By the Rev. A. D. Crake, B.A.
Crown ^fvo, js. 6d.
The Rival Heirs ; or, The Third and Last Chronicle
of .^scendune. By the Rev. A. D. Crake, B. A.
Crown Svo. 5j.
Allegories and Tales. By the Rev. w. E. Heygate, m.a.,
Rector of Brighstone.
With numerous Illustrations, Crown Svo. ys, 6d,
Stones of the Temple j or, Lessons from the Fabric
and Furniture of the Church. By Walter Field, M.A.,
F. S . A. , late Vicar of Godmersham,
With Illustrations. Royal i6ffto.
Stories from English History. By Louise Creighton,
author of "A First History of England," "Life of the Black
Prince," &c.
WA7EEL00 PLACE, LONDON.
EDUCATIONAL LIST, 37
BY y. HAMBLIN SMITH.
Elementary Algebra. SmaU $vo, p, WUhoui Answers^ 2s. td.
Key to Elementary Algebra. Crvwn Svo, 9s.
Exercises on Algebra. Smaii $vo, 2s, ed.
Arithmetic. Small ^0, y, 6d.
Key to Arithmetic. Craim &vo. gs.
Elements of Greometry. spuOI 2fvo, y. 6d.
Books I. and II., limp cloth, price is, 6d,, may be had separately.
Key to Elements of Geometry. Crmvn Uo. %s. 6d,
Trigonometry. Small Svo, ^. 6d.
Key to Trigonometry, croim ^vo. 7s. 6d.
Elementary Statics. SpmII ^0, y.
Elementary Hydrostatics. Smaii 8z/^, 3^.
Key to Elementary Statics and Hydrostatics. CroTvn
^0, 6s,
Book of Enunciations for Hamblin Smith's Geometry, Al-
gebra, Trigonometry, Statics, and Hydrostatics. Small Sfi^o, is.
An Introduction to the Study of Heat. Small ^0, y.
Latin Grammar, crawn ik/o, y. 6d,
Exercises on the Elementary Principles of Latin
Prose Composition. Crown Svo, y, 6d,
Key to Exercises on Latin Prose Composition.
Crown ^0, ^s.
An Elementary Greek Grammar. Cnmm 8zw, 4s, ed.
The Rudiments of English Grammar and Compo-
sition. Crown ZvOf 2s, 6d,
Notes on the Greek Text of the Acts of the Apostles.
Crown Swf, 4s, 6d,
Notes on the Greek Text of the Grospel of St. Mark.
Crown 2/vo, 2s. 6d.
WATERLOO PLACE, LONDON,
38 MESSRS. RIVINGTON'S LIST,
BY AR THUR SIDG WICK.
An Introduction to Greek Prose Composition.
Crown 9vo, 5^. A Key. ss,
A First Greek Writer. Crawn &». y. 6«/. ■ A Key. 51.
Cicero de AmiciUa. smaa Szv. 2s.
Homer's Iliad. Sma/I ^0, books I. and II. 2s, 6d, BOOK
XXL ij. 6d, BOOK XXIL is. 6d.
Scenes from Greek Plays, sma/i Svo, each is. 6d.
ARISTOPHANES : The Clouds, The Frogs, The Knii^ts, Phitus.
EURIPIDES : Iphigenia in Tanris, The Cyclops, Ion, Electra,
Alcestis, Bacchse, Hecuba.
BV GEORGE L. BENNETT.
First Latin Writer. Cnnm Svo. y. 6d. A Key. ss.
First Latin Exercises. Crtmm Svo. 2s. 6d.
First Latin Accidence. Craum fiuo. is. 6d.
Second Latin Writer. Cnmn Uo. y.ed. A Key. 55.
Elasy Latin Stories for Beginners, with Vocabularies and
Notes. Crown Bvo. 2s, 6d. A Key. ^s,
A Second Latin Reading Book. Crown ^0.
Selections from Caesar. The Gallic War. Sma// 8zv, 2x.
Selections from Vergil. st/Mi/ ^o, is. ed.
BY R. W. TAYLOR.
Xenophon'S Anabasis. Crmvn %vo. Books I. and n., 3Jf. 6e^;
III. and IV., 3J. dd.' Also separately, Book I., 2s, 6d, II., 2s.
Xenophon'S Agesilaus. Crawn $vo. 2s, 6d,
A Short Greek Syntax. Cnwu %vo. 9^.
Stories from Ovid in Elegiac Verse. Crawn Svo, y. 6d.
Stories from Ovid in Hexameter Verse, metamor-
PHOSES. CroTun ^o. 2s. 6d.
Scott's Lady of the Lake. Formmg a Volume of the " Englidi
School Classics." Small 2m, 2s. ; or in Three Parts, each 9^
WATERLOO PLACE, LONDON.
INDEX
PAGE
Abbott (E.), Arnold's Greek Prose 23
— Elements of Greek Accidence . 23
— Hellenica . . . . . 27
— Selections from Lucian . . 24
— and Mansfield (E.D.), Primer of
Greek Grammar ... ax
Acland (A.), Political Hist, of Eng. 10
Adams (W.), Allegories ... 36
Alford n^ean) Greek Testament . 29
Aristophanes 35, 28
Aristotle's Ethics .... 38
— Organon. By J. R. Magrath . 28
Arnold ^T. K.) Cornelius Nepos . ao
— Crusius' Homeric Lexicon . . 26
— Demosthenes .... 27
-— Eclogae Ovidianae ... 18
— Eng. Greek Lexicon . . 29
— First French Book and Key . 34
— First German Book and Key . 33
— First Greek Book and Key . 22
; revised by F. D. Morice . 22
— F'irst Hebrew \io6k. and Key . 35
— First Verse Book and Key . 16
— (Jreek Accidence ... 22
— (ireek Prose Comp. and Key . 92
revised by E. Abbott . . 23
— Henry's First I^tin and Key . 16
rexised by C. G. Gepp . . 16
— Homer's Iliad .... 26
— I^tin Prose Comp. and Key . 17
revised by G. G. Bradley . 17
— Mad Nag's Greek Syntax . . 22
— Sophocles '28
Bakkk (W.), Manual of Devotion .
Harbier (P.), French Reader .
Barrett (W. A.), Chorister's Guide .
— Form and Instrumentation
Belcher (H.), Livy, Book II. .
Bennett (G. L.X Caisar's Gallic War
— Easy Latin Stories and Key
— First Latin Exercises .
— First Latin Writer and Key
— Latin Accidence ....
— Second Latin Writer and Key .
— Vergil, Selections from
— Works by
Bigg (C), I^tin Prose Exercises .
— Thucydides, Books 1. 11. .
Blunt (J. H.), Household Theology
— Ke>-s to Christian Knowledge .
}k>wen (E. E.), Campaigns of Napoleon
Bradl«y(G. G.), AmolcTs Latin Prose
Bridge (C.XFrcndi Literature
Bright CJ. F.). History of England .
Buuding Construction. Notes on .
Ikirton (J.), English Grammar
CiSSAR . . . . . 15, z6, x8
Calvert (E.), Selectioiis from Livy . ao
Can- (A.), Notes on St. Luke . . ag
3a
34
35
35
19
15
15
14
14
14
14
17
38
18
28
3a
31
34
17
Catena Gassiccn^m
Cicero de Amicitia . . ' .
Clarke (A. D.), Examination Papers
Cornelius Nepos. By T. K. Aniold
Cornish (F. Wjl, Oliver Cromwell .
Crake (A. D.), History of the Church
— Edwy the Fair . , . .
— Alfgar the Dane
— Ki\'al Heirs . . . .
Creighton {LX First Hist, of Eng.
-«- Historical Biographies
— Stories from Eng. Hist.
Crusius (G. CX Homeric Lexicon .
Curteis (A. M.), The Roman Empire
Dallin (T.X Materials and Models
Davys (Bishop), History of England
Dawe (C. J. S.), Latin Exercise Bk
— Study of Church Catediism
Demosthenes ....
II
5
English School-Classics
Euripides, Scenes from .
FiKF.i) (W.), Stones of the Temple
Firman (F. B.JL Zeugma .
Fradersdorif, Eng.-Greek Lex.
Gantillon(P.J.F.), Exam. Papers
Gedge (I. W.), Com. to Braver Book
Gepp (C. G.), AmoW's Henry's First
I^tin Book ....
— I^tin ElegUc Verse .
r- I..atin-English Dictionary .
— VirgU
Girdlestone (W. H.), Arithmetic .
Goethe's Faust ....
Goolden(W. T.), Intra to Chemistry
Goulbum (Dean), Confinnati<Mi ,
Green (A. H .), Geology for Student!
— (W. C^ Aristophanes
Grenfell (E. FX German Exercises
Gross (E. J.), Algebra, Part 11. ' .
— Kinematics and Kinetics .
Hardy (E. G.). Antiq. of Greece . ay
Harrison (J. E.) Myths of the Odyssey 27
HaufTs Stories, Selecticms from . 33
Heatley (H. R.), (^datim . • ■ X5
Hellenica, Essays . . . . 37
Herodotus, Stones frtMn, Phillpotts . 34
— By H. G. Woods ... 37
Hertz (H. A.), Short Readings . 5
Heslop (G. H.), Demosthenes . 27
Hcygate (W. E.), Allegories . . 36
Historical Biographies ... 10
Historical Handbooks ... 9
Holmes (A0» Demosthenes . . aj
— Rules of^ Latin Proncuiciation . 15
Homer's Iliad a6
Horace. By J. M. Marshall . ao
loPHON a4
Isocrates. By J. E. Sandys . 26
PAGB
!S
»3
ao
zo
so
36
36
36
8
xo
3«
a6
»9»a4
xo
X5
32
37
6,7
85
36
•4
*9
»9ia4
3x
x6
x6
19
17
13
33
xz
3x
XX
a8
33
xa
xa
s • * -
INDEX,
\
PAGB
BBB (R. C.)} Sophocles ... 28
ennings (A. C), Ecclesia Anglicana 32
uveiuu. By G. A. Simcox . . ao
Kkys to Chbistian Knowlbdgb .
— List of
Kitchener (F. A.), A Year's Botany
Kingdon (H. N.), Gradatim .
Latin Tbxt Books
La Fontaine's Pabl)£k By P. Smith
Lang (L. B),Geography for Beginners
)n(j.),/
Laun (Van, H), French Selections
, At Home and Abroad
Laughtor
Laun (Vaw, **/, a^w»^,u wwt«^^...v
E^sin^'s Fables. By F. Storr
Libraries, Books suitable for .
Livyi Selections from
Lucian, Selections from .
Macbbth (J.), Church Principles
Madvig's Greek. Syntax .
MagrathCJ. R.), Aristotle's Organon
Mann (J. SO* Aitiquities of Greece
Mansfield (£. D.), Latin Sentence .
— Primer of Greek Syntax .^
Manuals of Religious Instruction .
Marshall (J. M.), Horati Opora
Merryweather (J. H.), Caesar .
Moberly(C.£.), Alexander the Great
— Geography
— Shakspere's Play; . .
— Xen(^hon's Memorabilia .
Moore (£.), i^ristotle's Ethics .
Moore cE. H.), Greek Method
Morice(F. D.), Stories in Attic Greek
Morshead (E.D.), Goethe's Faust .
Mullins (W. E.), HauflTs Stories .
Napoleon's Campaigns
Norris CF. P.), New Testament
— Confirmation .
— Keys to Christian Knowledge .
-^ Rudiments of Theology
OviDiANiS Eclogae. By Arnold
Ovid, Stories from. By R. W. Taylor
Papillon (T. L.), Terenti Comoediae
Pearson TC. H^, English History
Percival (J.), Helps for School Life
Persius. By A. Pretor .
Phillpo'tts (J. S.). Homer's Iliad
— Shakspere's 1 empest
— Stories from Herodotus
PoweU (F. York)j English History
Prefepr (A.), Persii Satirae
Priestland (E.), Greek Prepositions
Prizes, Books suitable for
RANSOMK(C0f Political Hist of Eng
— History of^tlfi Romans
Reynolds (S. H.% IBad of Homer
Richardson (G.), Conic Sections
Rigg (A.), Intro, to Chemistry
Ritchie (F.), First Steps in Latin
— Practiced Greek Method .
Rivington's Mathematical Series
Rogers (J. E. T.), Aristotle's Ethics
12,
31
4
XI
XS
16
34
35
35
34
33
36
20
24
32
22
28
27
19
21
31
20
18
25
35
5
25
28
21
25
33
33
34
32
31
31
32
18
18
20
9
31
20
26
5
24
8
20
23
10
9
26
12
II
15
21
13
28
PAGB
Sanderson (L.), Zeugnoa . . 24
Sandys Q. £.), Isocratis Orationes 26
Sargent Q.), Materials and Models . 19, 34
Schoemann s Antiquities of Greece 37
Shakspere's Plays . ... 5
Sidgwick (A.), Cicero de Amicitia . x8
— First Greek Writer . v • • *«
— Greek Prose Compositidb . . m
— Homer's Iliad . . . . a6
— Scenes from Greek Plays . .as
— Works by . . . . . 38
Simcox (G. A.), Juvenalis Satirae . so
— Thucydides . •. •. • "S
— (W. H.), Taciti Historise , ao
Smith (J. Hamblin), The Acts . ag
— Algebra and Key . . . xa
-^ Algebra, Exercises on . . za
— Arithmetic and Key . . . xa
— Book of .jEnnnci^tioiis . . 13
— English Gramniar ... . 5
— Geometry and Key . . . .^.13
— Greek Graminar . . "t* . "."J13
— Heat, The Study of . . . X3
— Hydrostatics dmd Key . . xa
— Latin Gramnjar . . . . z6
Prose Composition and Key . x8
— Statics and Key . . ... xa
— St. Mark's Gospel ... 29
— Trigonometry and Key . . la
— Works by 37
— f P. Bowden), La Fontaine's Fables 34
— (P. v.), English Institutions . 9
— iR. Prowde), Latin Prose Ex. . 15
Sophocles ..... b8
Storr (F.), iEneid of Vergil . . 17
— Greek Veri)s^ 33
— HauflTs Sfones. •'.-''•, . . 33
I — Lessing's FalJte$'' • • • 33
Tacitus. By W. H. Simcox . . ao
I Tancock (C. C), Caesar . . x8
, Taylor (R. W.), Short Greek Syntax
i — Stones from Ovid
— Xenophon's Agesilaus
— Xenophon's Anabasis
— Works by . •. .
, Terence. By T. L. Papillon .
. Tidmarsh (W.) English Grammar .
Thucydides .*....
; Turner (E. J.) Goethe's Faust
as
as
38
ao
5
aS
33
I Vecquerav (J.), German Accidence 33
Vergil t6, X7
Waite (R.), Duke of Wellington
Way of Life ....
Whitelaw, Shakspere's Coriolanus
Willert (P. F.), Reign of Lewis XI
Wilson's Modem English Law
Woods (H. G.), Herodoti Historia
Wordsworth (Bp.), Greek Testsunent
Wormell, Principles of I)ynamics
Worthington's Practical Physics
•*.
Xbnophon
xo
3«
5
9
9
•7
•9
X3
XX
•5
RIVINGTONS' EDUCATIONAL LIST
A rnolcPs L atin Prose
Compontum, By G. 6. Bradley.
5*.
[The original Edition is still on sale.]
ArnolcTs Henrys First
LoHn Book, By C G. Gbpp. y.
[The original Edition is still on sale.]
First Latin Writer. By
G. L. Bbnnbtt. 3X. 6d.
Or separately —
First Latin Exercises, as.ed,
Latin Accidence. \s. 6d.
Second Latin Writer.
By G. L. Bbnnbtt. ^. 6d.
Easy Latin Stories for
Beginners. By G. L. Bennett.
Selections from Ccesar.
By G. In Bennett. 2J.
Selections from Vergil,
By G. L. Bennett, is. Gd.
Virgil Georgics, Book iv.
By C. G. Gepp. i*. (xL
CcBsar de Bello Gallico,
Books I — III. By J. Merry-
weather and C Tancock. 3^. td.
Book I. separately, u.
The Beginne/s Latin
Exercise Book. By C. J. S. Dawb.
\s, 6d,
First Steps in Latin, By
F. Ritchie, u. 6d.
Gradatim, An Easy Latin Trans-
lation Book. By H. Heatlby and
H. Kingdon. xs. 6d.
Arnolds Greek Prose
Composition. By E. Abbott. 31. 6d.
[The original Edition is still on sale.]
A Primer of Greek
Grammar, By E. Abbott and E.
D. Mansfield. 3;. td.
Or separately —
Syntax, xs. &/.
Accidence, at. td.
A Practical Greek Me^
thod for Beginners. Thb Simflb
Sentencb. By F. Ritchie and E.
H. MooRB. 3X. td.
Stories in Attic Greek,
By F. D. MoRiCB. jf. td.
A First Greek Writer,
By A. SiDGWiCK. 3X. &/.
An Introduction to Greek
Prose Composition. By A. Sidg-
WICK. s*.
Homers Iliad. By a. sidg-
WICK.
Books I. and II. 2f. 6<^
Book XXI. 1*. td.
Book XXII. I*. &£
Tlie Anabasis of Xeno-
plion. By R. W. Taylor.
Books I. and II. 31. 6d.
Or separately. Book I., 2s. 6d, ;
Book II., aj.
Books III. and IV. 3^. 6d.
Xe7iophoris Agesilaus,
By R. W. Taylor, sj. 6rf.
Stories from Ovid in
Elegiac Verse. By R. W. Taylor.
3*. td.
Stories from Ovid in
Hexameter Verse. By R. W. Tay-
lor. 2f . td.
•.■-•. ■>.•.
Waterloo Place, Pall Mall, London.
RIVINGTONS' EDUCATIONAL LIST
Select Plays of Sluxkspere,
Rugby Edition.
By the Rev. C. £. Mobbrly.
AS YOU LIKE IT. «.
MACBETH. M.
HAMLET, su. 6^
KING LEAR. m. 6rf.
ROMEO AND JULIET, a*.
KING HENRY THE FIFTH, aj.
MIDSUMMER NIGHT'S
DREAM, u.
By R. Whitklaw.
CORIOLANUS. 9J. &/.
By J. S. Phillpotts.
THE TEMPEST, m.
A History of England.
By the Rev. J. F. Bright.
Period I.— MsDiiEVAL Monarchy :
A.D. 440—1^5. 4J. 6d.
Period II.— Personal Monarchy;
A.D. 1485 — 1688. 5*.
Period 111.— Constitutional Mon-
archy: A.D. Z689 — 1837. 7X. 6<i.
Historical Biographies,
By the Rev. M. Crbighton.
SIMON DE MONTFORT. ax. 6d.
THE BLACK PRINCE, aj. 6d.
SIR WALTER RALEGH. 3J.
DUKE OF WELLINGTON. 2s,6d.
DUKE OF MARLBOROUGH.
IVER CROMWELL, y. 6rf.
A Handbook in Outline
of EttgUsh History to 1881. By
Arthur H. D. Acland and Cyril
Ransoms, ds.
A First History of Eng-
land, By LouiSB Crbighton. With
Illustrations, us. 6d,
Army and Civil Service
Examination Papers in A rithmetic.
By A. Dawson Clarkb. 31. td.
SJiort Readings in Eng-
lish Poetry. By H. A. Hbrtz.
ax. &/.
Modem Geography^ for
the Use of Schools. By the Rev. C.
E. Mobbrly.
Part I.— Northern Europb. ax. td.
Part II.— Southern Europb. ax. &/.
A Geography for Begin-
ners. By L. B. Lang.
THE BRITISH EMPIRE, ax. 6d.
Part I. — The British Isles, ix. 6d,
Part II.— Thb British Posses-
sions, zx. 6d,
A Practical E^tglisk
Grammar. By W. Tidmarsh.
2X. 6</.
A Graduated French
Reader. By Paul Barrier, ax.
La Fontaines Fables,
Books I. and II. By the Rev. P.
Bowdbn-Smith. ax.
^^ *»^ ^\^r r'^s"
Goethe's Faust. By e. j.
Turner, and E. D. A. Morshead.
7X. 6d.
Lessing's Fables. By f.
Storr. 2x. 6d.
Selections from HaufTs
stories. By W. E. Mullins and F.
Storr. 4X. 6d,
Also separately —
KALIF STORK AND THE
PHANTOM CREW. ax.
A Gerfnan Accidence,
By J. W. J. Vecqueray. 3X. td.
German Exercises, Adapted
to the above. By E. F. Grbn-
pbll. Part I. ax. Part II. ax. td.
Waterloo Place, Pall Mall, London.
4
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11